ThreeMode 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.
INDEX
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 nmode components analysis. Psychometrika,
51, 269275.
As an extension of Lastovicka's fourmode components analysis an nmode
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 nmode
extension of the threemode 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 pseudoternaire par la
méthode d'analyse binaire de Faverge. Le Travail Humain, 38,
287300.
Proposal to string out a threemode matrix in one of three
ways with the object to perform on the resultant (twomode)
matrix a variant of the singular value decomposition (called
the binary method of Faverge). One of the stringingout
proposals is identical to forming a combinationmode (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,
1112511132.
Structural features of the single and mixed micellar systems of sodium
dodecyl sulfate (SDS) and dodecyltrimethylammonium bromide (DTAB) were
characterized using the fluorescence probe 6propionyl2(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 threemode factor analysis technique to resolve the coincident emission
from Prodan in multiple microenvironments of single and mixed micelle
systems.

Kelly, E.F., Lenz, J.E., Franaszczuk, P.J., & Truong, Y.K.
(1997).
A general statistical framework for
frequencydomain analysis of EEG topographic
structure.
Computers and Biomedical Research, 30, 129164.**
A wide variety of rhythmic electrophysiological phenomenaincluding driven,
induced, and endogenous activities of cortical neuronal masseslend themselves
naturally to analysis using frequencydomain 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 sourcerecovery 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 arisenot only in the spatial distribution of
the activity, but also in its frequencydependent betweenchannel covariance
structureas 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 singlechannel spectral measures
(and measures of betweenchannel 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, 433460.
Five extensions of the classical twoset 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, 161191.
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 realdata examples, are
presented.

Kettenring, J.R. (1983a).
A case study in data analysis. Proceedings of Symposia in
Applied Mathematics, 28, 105139.
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. 283297). Amsterdam:
NorthHolland.
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.
For
twoway 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 twoway interaction terms estimated from the data.
Probability plots of the
singular values from this decomposition vs. computergenerated null
values provide an
effective way of informally implementing this approach.
Extensions of
these ideas to deal
with threeway interactions are also feasible. In some contexts, it makes
sense to treat
these interactions as if they were layers of twoway interactions and to
apply methods which
exploit this viewpoint. More generally, an extended form of the singular
value decomposition,
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 twoway
case. Each of these
approaches could be broadened to treat higherway interactions as
well.
This paper
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 "AngloSaxon" and "French" threemode methods.
Statistique et Analyse des Données, 13, 14
32.
Seven methods for the analysis of threemode 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
comparison of
french methods and anglosaxon methods.

Kiers, H.A.L. (1989a).
INDSCAL for the analysis of categorical data. In R. Coppi & S.
Bolasco (Eds.), Multiway data analysis (pp. 155167).
Amsterdam: Elsevier.
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
described.

Kiers, H.A.L. (1989c).
An alternating least squares algorithm for fitting the two and three
way DEDICOM model and the idioscal model. Psychometrika,
54, 515521.
The DEDICOM nodel is a model for representing asymmetric relations
among
a set of objects by means of a set of coordinates for the objects on a
limited
number of dimensions. The present paper offers an alternating least
squares
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.
An
algorithm for fitting this threeway DEDICOM model an algorithm is
developed
for fitting the IDIOSCAL model in the least squares sense.

Kiers, H.A.L. (1989d).
A computational shortcut for INDSCAL with orthonormality
constraints on positive semidefinite matrices of low rank.
Computational Statistics Quarterly, 5, 119135.
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 semidefinite (p.s.d.) matrices of low
rank, computations
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 lowrank p.s.d.
matrices is that of INDORT on a set of quantification matrices for
qualitative and/or
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
to quantification
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
variables.

Kiers, H.A.L. (1990a).
Procrustespc v2.0: Een programma voor gegeneraliseerde
Procrustesanalyse. [Procrustespc v2.0: A program for generalized
Procrustesanalysis.] Kwantitatieve Methoden, 11,
177188.
In this paper a description and evaluation are given of a pcprogram for
generalized
Procrustesanalysis. After a short discussion of the aim of generalized
Procrustesanalysis,
the possibilities and restrictions of the program are discussed. The
operation and
userfriendliness of the program are looked into using several simple
testanalyses.

Kiers, H.A.L. (1991a).
Hierarchical relations among threeway methods.
Psychometrika, 56, 449470.
Several methods have been developed for the analysis of a mixture of
qualitative and
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,
197212.
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 quantitaive variables, called INDOMIX, is shown to
construct
component scores (without rotational freedom) maximizing the quartimax
criterion
over all 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. (1992).
TUCKALS core rotations and constrained TUCKALS modelling.
Statistica Applicata, 4, 659667.
TUCKALS3 is a method for analyzing a threeway data set. The method
decomposes the data into component matrices for all three sets of
components.
The solution for the component matrices and the core array are determined
up to
a (possibly oblique) rotation only. In the present paper some procedures
are
proposed for rotating the solution such that the core becomes optimally
simple in
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
specified
places. An algorithm for this procedure is sketched and the procedure is
illustrated
by means of an example.

Kiers, H.A.L. (1993a).
An alternating least squares algorithm for PARAFAC2 and threeway
DEDICOM. Computational Statistics and Data Analysis, 16,
103118.
PARAFAC2 is a method for analyzing threeway data consisting of
symmetric
frontal slices. Threeway DEDICOM can be considered a generalization of
PARAFAC2 in that it fits essentially the same model to threeway data
consisting
of square frontal slices that may be asymmetric. In the present paper, an
alternating least squares algorithm is developed for threeway DEDICOM,
and an
algorithm for PARAFAC2 is derived from it. The performance of the
algorithms
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
Directions 2
(pp. 6786). 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
analysis (PCCA),
INDOMIX, 'Varimax Optimization' and INDSCAL applied to quantification
matrices for
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
and Varimax
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
set.

Kiers, H.A.L. (1993c).
Handling ordinal variables in threeway analysis of
quantification matrices for variables of mixed measurement levels.
British Journal of Mathematical and Statistical Psychology,
46, 135152.
For the analysis of variables of mixed measurement levels a class of
methods can be used that
is based on threeway analysis of quantification matrices for nominal or
quantitative
variables. This class of methods incorporated some wellknown techniques
but also offers a
series of interesting new alternatives for the analysis of nominal or
quantitative variables.
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
these variables.
Algorithms for this optimal scaling procedure are developed, and the
optimal scaling
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
threeway analysis of two types of quantification matrices.
Applied Stochastic Models and Data Analysis, 9, 301317.
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 coordinates 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 quadraticforms and some generalizations.
Psychometrika, 60, 221245.
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), H1(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, H2(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 Ck 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,
297310.
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).
Threemode orthomax rotation.
Psychometrika, 62, 579598.
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 threemode pca, rotational freedom of
the so called core (a threeway 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 (twoway) 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 byproduct, application of the
threemode orthomax procedures to twoway arrays
is shown to reveal interesting relations with and
interpretations of existing twoway simple
structure rotation techniques.

Kiers, H.A.L. (1997c).
Weighted least squares fitting using ordinary
least squares algorithms.
Psychometrika, 62, 251266.
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
imputation.

Kiers, H. A. L. (1997d).
Discrimination by means of components that are orthogonal in the data space.
Jounal of Chemometrics, 11, 533545.
Krzanowski (J. Chemometrics, 9, 509 (1995)) proposed a method for
obtaining socalled 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 threemode factor analysis: Constrained
threemode 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. 563574). Tokyo: Springer
Verlag.
A review is presented of some recent developments in threemode factor
analysis, that are all
aimed at reducing the difficulties in interpreting threemode factor
analysis solutions.
First, variants of threemode 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
structure are
discussed and related to twoway simple structure rotation techniques. In
the concluding
section, new perspectives for simplification of the interpretation of
threemode factor
analysis solutions are discussed.

Kiers, H.A.L. (1998b)
An overview of threeway analysis and some recent developments.
In A. Rizzi, M. Vichi
& H.H. Bock (Eds.), Advances in data science and
classification (pp. 593602). Berlin: Springer.
A brief overview is presented of techniques for the analysis of
threeway 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 threeway models are distinguished
with respect to their uniqueness. Next, recent approaches are
discussed for eliminating nonuniqueness by imposing constraints
and rotation to simple structure.

Kiers, H.A.L.(1998c).
A threestep algorithm for CANDECOMP/PARAFAC analysis of large
data sets with multicollinearity.
Journal of Chemometrics, 12, 155171.
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
timeconsuming. The present paper describes a threestep
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.

Kiers, H.A.L.(1998d).
Joint orthomax rotation of the core and
component matrices
resulting from threemode principal components analysis. Journal of
Classification,
15, 245263.
The analysis of a threeway data set using threemode principal
components analysis yields
component matrices for all three modes of the data, and a threeway array
called the core,
which relates the components for the different modes to each other. To
exploit rotational
freedom in the model, one may rotate the core array (over all three
modes) to an optimally
simple form, for instance by threemode 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
the component
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
reasonably simple.

Kiers, H.A.L.(1998e).
Threeway SIMPLIMAX for oblique rotation of the threemode factor analysis core
to simple structure.
Computational Statistics & Data Analysis, 28, 307324.
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 threeway arrays: Threeway SIMPLIMAX finds oblique
simple structure
rotations of the core matrix that results from a threemode factor
analysis. Specifically,
threeway SIMPLIMAX minimizes the sum of the m smallest elements
of the rotated core
array. An algorithm for threeway 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 Nmode factor analysis, and for rotation
over a subset of the
modes of an Nmode core array.

Kiers, H.A.L. (2000a).
Some procedures for displaying results from threeway methods. Journal of
Chemometrics, 14, 151170.
Threeway Tucker analysis and CANDECOMP/PARAFAC are popular methods for the
analysis of threeway 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 threeway methods. Not all of these optimally correspond
to the actual approximation of the data furnished by the threeway method at
hand. Next it is described how plotting procedures can be designed that do
correspond exactly to the lowdimensional description of the data by means of
the threeway 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 threedimensional 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
appendix.

Kiers, H.A.L. (2000b).
Towards a standardized notation and terminology in multiway analysis.
Journal of Chemometrics, 14, 105122.
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 higherway 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
made.

Kiers, H. A. L. (2004).
Bootstrap confidence intervals for threeway methods.
Journal of Chemometrics, 18, 2236.
Results from exploratory threeway 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 nonuniqueness 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 canonicalanalysis 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 innerproduct
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
positive semidefinite.

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,
119125.
Recently, Timmerman and Kiers proposed an effective procedure for choosing
the numbers of components in Tucker3 analysis, a kind of component analysis
of threeway data. The procedure, however, is rather timeconsuming, relying
on very many complete Tucker3 analyses. Here, an alternative procedure is
proposed, which basically relies on a single, quick analysis of the threeway
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 threeway principal component and related multilinear
models. Chemometrics and Intelligent Laboratory Systems, 36,
3140.
Multilinear analysis methods such as component (and threeway 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
(threemode) PCA algorithm to a small (threeway) 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 (threemode) PCA algorithms to
the small array. Finally, it is shown how the latter approach for speed
improvement can also be used for other threeway models and analysis
methods (e.g., PARAFAC/CANDECOMP and constrained threemode PCA).

Kiers, H.A.L., & Krijnen, W.P. (1991).
An efficient algorithm for PARAFAC of threeway data with large
numbers of observation units. Psychometrika, 56,
147152.
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
during the
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, 415422
A modification of the TUCKALS3 algorithm is proposed that handles
threeway
arrays of order I x J x K for any I. When I is much larger than JK, the
modified
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
additional
feature that execution speed is higher, the modified algorithm is highly
suitable
for use on personal computers.

Kiers, H.A.L., & Marchetti, G.M. (1994).
Handling large numbers of observation units in
threeway methods for the analysis of qualitative and
quantitative twoway data.
Computational statistics, 9, 4164.
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 threeway
methods like indscal, idioscal and tuckals3. When the number of
observation units (objects) is
large, algorithms for indscal, idioscal and tuckals3 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 threeway
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
for nominal
variables.

Kiers, H.A.L., & Smilde, A.K. (1995).
Some theoretical results on secondorder calibration
methods for data with and without rank overlap.
Journal of Chemometrics, 9, 179195.
Gram, a method for secondorder 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 (rankone) 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 secondorder
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
secondorder 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 threemode factor analysis as a tool for parameter
estimation with secondorder instrumental data Journal of
Chemometrics, 12, 125147.
Threemode factor analysis models are often used in exploratory
analysis of threeway data. However, in some situation it is a
priori known that a particular constraint threemode 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
assessed.

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, 467473.
Millsap and Meredith (1988) have developed a generalization of
principal
components analysis for the simultaneous analysis of a number of
variables
observed in several populations or on several occasions. The algorithm
they
provide has some disadvantages. The present paper offers two alternating
least
squares algorithms for their method, suitable for small and large data
sets,
respectively. Lower and upper bounds are given for the loss function to
be
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,
13, 275294.
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 threeway 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 threeway data in situations where the
PARAFAC1 model is too restricted. Usually the PARAFAC2 model is fitted to a set
of matrices with crossproducts 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 higherway 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
crossproduct matrices only.

Kiers H.A.L., Ten Berge, J.M.F., & Rocci, R. (1997).
Uniqueness of threemode factor analysis models with sparse cores:
The 3×3×3 case.
Psychometrika, 62, 349374.
Threemode PCA and PARAFAC are methods to describe threeway data.
The threemode PCA model also uses a threeway 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 threemode PCA model has nonunique
components and it seems hard to choose among all possible
solutions. The present paper introduces a class of threemode PCA
models in between threemode 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).
Threeway component analysis: Principles and illustrative application.
Psychological Methods, 6, 84110.
Threeway component analysis techniques are designed for descriptive
analysis of 3way data, for example, when data are collected on individuals,
in different settings, and on different measures. Such techniques summarize
all information in a 3way data set by summarizing, for each way of the 3way
data set, the associated entities through a few components and describing
the relations between these components. In this article, 3mode 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
3way data by this technique. The complete process is illustrated with a
detailed description of the analysis of an empirical 3way data set.

Kjerulff, K. & Wiggins, N.H. (1976).
Graduate student styles
for coping
with stressful situations. Journal of Educational Psychology,
68, 247254.
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
variables.

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. 319342).
Mahwah, NJ: Erlbaum.
Network models aim at representing proximity data by means of the minimum
pathlength function
of connected and weighted graphs. Fundamental representation and
uniqueness results underlying
network models as psychological representations of stimuli, given both
ordinalscale as well
as intervalscale 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 twoway
and threeway 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
proximity data.

Knobloch, E.M. (1972).
Einschätzung von leistungsrelevanten
Begriffen.
Unpublished master thesis, University of Hamburg, Hamburg,
FRG., 1972.

Kohler, A. (1980).
Das TrimodProgrammSystem (TRIPSY) zur
Berechnung
der dreimodalen Faktorenanalyse nach Orlik (manuscript
in preparation).
Description of a computer program implementing Orlik's (1981)
Summax model.

Kofides, E., & Regalia, P. A. (2002).
On the best rank1 approximation of higherorder supersymmetric tensors.
Siam Journal of Matrix Analysis and Applications,23, 863884.
Recently the problem of determining the best,in the leastsquares sense,rank1
approximation to a higherorder tensor was studied and an iterative method that extends the well
known power method for matriceswasproposed for itssolution.Thishigherorder 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 higherorder 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,
23, 243255.
We explore the orthogonal decomposition of tensors (also known as multidimensional
arrays or nway 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 EckartYoung SVD
approximation theorem by Leibovici and Sabatier.

Kolda, T. G. (2003).
A couterexample to the possibility of an extension of the eckartyoung lowrank
approximation theorem for the orthogonal rank tensor decomposition.
Siam Journal of Matrix Analysis and Applications,24, 762767.
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 threeway decomposition to the analysis of heteronuclear NMR
relaxation data. Journal of Biomolecular NMR,21, 263268.
MUNIN (Multidimensional NMR Spectra Interpretation), a recently introduced
approach exploiting the mathematical concept of threeway decomposition, is proposed for
separation and quantitative relaxation measurements of strongly overlapped resonances in
sets of heteronuclear twodimensional spectra that result from typical relaxation experiments.
The approach is general and may also be applied to sets of twodimensional spectra with arbitrary
modulation along the third dimension (e.g., Jcoupling, 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, 138146.
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 batchtobatch 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
batchtobatch variability is due to reactor
temperature variations arising from disturbances
in the heating system and other heattransfer
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, 321.
Multivariate statistical methods for the analysis,
monitoring and diagnosis of process operating
performance are becoming more important because of
the availability of online 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 t2 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), multiblock
pls and multiway 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, 277284.
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 setup 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).
Driemodale faktoranalyse.
Programmabeschrijving
(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, 17871795.
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 threeway arrays by constrained PARAFAC
methods. Leiden: DSWO Press.
In this book  a companion volume to Kroonenberg's Threemode Principal
Component
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
methods are
introduced that allow for easier interpretations. The first one employs
constant relative
importances of the components across occasions, the second one determines
contrastfree
components, and the third one determines components that correspond to
nonoverlapping
clusters of variables. The usefulness of the constrained PARAFAC methods
is illustrated by
various analyses of empirical threeway data, and by simulations.

Krijnen, W.P. (1993).
Noncontrast 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. 237241). Stuttgart
and New York: Gustav Fischer Verlag.
If a threeway 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 noncontrast components. In case one finds contrast components
with the PARAFAC method, the uniqueness property prevents using a rotation to find
noncontrast components. A restricted PARAFAC method is presented that identifies
noncontrast components that optimally represents the variables. In case the fit
of the noncontrast components is close to the fit of he ordinary PARAFAC components
the noncontrast 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. 165177). 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
nonoverlapping clusters. In addition a second variant of PARAFAC is
proposed that identifies an optimal PARAFAC representation which has
components corresponding to nonoverlapping 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, 299306.
The method for analyzing threeway 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 lossfunction
values, and that Weighted PCA attains these values using fewer floating
point operations.

Krijnen, W.P., & Ten Berge, J.M.F. (1991).
Contrastvrije oplossingen van het CANDECOMP/PARAFACmodel
[Contrastfree solutions of the CANDECOMP/PARAFACmodel].
Kwantitatieve Methoden, 12, 8796.
The CANDECOMP/PARAFAC decomposition of threeway 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 contrastfree loadings.
CANDECOMP/PARAFAC is
indeed able to produce contrast components; the lack of rotational freedom
is a problem in
that case. The solution can be found by imposing nonnegativity
constraints. It turns out that
with a minimal loss, constrastfree 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, 295305.
A set of nonnegativity correlated variables, referred to as positive
manifold data,
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
correlate positively
with all components, thus displaying positive manifold. However,
this phenomenon is
critically dependent on the freedom of rotation, as is evident from the
unrotated loadings.
That is, although the first principal component is without contrast (which
means that all
variables correlate either positively or negatively with the first
component), subsequent
components have mixtures of positive and negative loadings  which means
that positive
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
positive manifold
solutions. For cases in which PARAFAC does not produce a positive manifold
solution, a
constrained PARAFAC method is offered that restores positive manifold by
introducing
nonnegativity constraints. Thus, noncontrast PARAFAC components can be
found that explain
only a negligible amount of variance less than the PARAFAC components.
These noncontrast
components cannot be degenerate and cannot be partially unique in the
traditional sense.

KrolakSchwerdt, S. (1991a).
Modelle der dreimodalen Faktorenanalyse. Frankfurt am
Main: Peter Lang.
Contents:
1. Introduction.
2. From the covariation chart to threemode factor analysis: A historic
milestone.
3. Threemode factor analytical models: Basic equations, algorithms and
interpretations.
4. Formal characteristics and specifics of the threemode factor analytical
model: A classification.
5. Coherence and theoretical relations between threemode factor analytical
models.
6. Application of the threemode factor analytical models: Preliminary
remarks.
7. Reanalysis of a modelexperiment by Orlik concerning the psychophysics of
polarity profiles.
8. Dimensions of colour perception: A comparison between MDSmodels and
individual differences structureanalysis.
9. Different types of sorting techniques: Rosenberg and Kim's study of
methods.
10. Osgood and Luria: A case of multiple personalities  an empirical comparison
of threemode data models.
11. Discussion.

KrolakSchwerdt, S. (1991b).
Modelle der dreimodalen Faktorenanalyse: Formale Eigenschaften,
theoretische Zusammenhänge und ihre Implikationen für das
Konzept individueller Differenzen. Psychologische
Beiträge, 33, 314346.
The present paper is concerned with methods of threemode factor analysis to
obtain a dimensional representation of threeway 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) threemode 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.

KrolakSchwerdt, S., Orlik, P., & Ganter, B. (1994).
TRIPAT: A model for analyzing threemode binary data. In H.H.
Bock, W. Lenski, & M.M. Richter (Eds.), Information Systems
and Data Analysis: ProspectsFoundationsApplications (pp.
298307). Berlin: Springer.
A discrete, categorical model is presented for threemode (conditions by
objects by
attributes) data arrays with binary entries x_{ijk} element
of {0,1}.
Basically, the model attempts a simultaneous classification of the
elements of the three modes
in a number of common clusters. Clusters are defined by threemode
submatrices of maximum size
with entries x_{ijk} = 1. In performing a discrete
representation of the data
structure, the model may be classified as a nonhierarchical clustering
procedure. It involves
a reorganization of the data array such that the final clustering solution
is interpreted
directly on the data, and it allows for overlapping as well as
nonoverlapping clusters. The
method is similar to threemode 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
three
mode principal component analysis. WEPreeks, WR 816RP, Vakgroep
W.E.P., University of Leiden, Leiden, The Netherlands,
1979(a).
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 threemode
principal
component analysis. WEPreeks, WR 8121EX, 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 threemode
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, WR8135RP, 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 threemode principal
component analysis. Kwantitatieve Methoden, 3, 6594.
After a relatively nontechnical account of threemode component analysis
of threeway 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 threemode principal component analysis may be used to
unravel complex
relationships.

Kroonenberg, P.M. (1983a).
Annotated bibliography of threemode factor analysis. British
Journal of Mathematical and Statistical Psychology, 36,
81113.
Published and unpublished theoretical and applied papers on threemode
principal component
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
treated.

Kroonenberg, P.M. (1983b).
Correlational structure of the subtests of the SnijdersOomen
nonverbal intelligence scale. Kwantitatieve Methoden,
4, 4051.
Using threemode 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).
ThreeMode Principal Component Analysis: Theory and
Applications. Leiden: DSWO Press.
Contents:
1. Preliminaries
2. Survey
Part I: THEORY
3. Models
4. Methods and algorithms
5. Transformations of core matrices
Part II: THEORY FOR APPLICATIONS
6. Scaling and interpretations
7. Residuals
Part III: APPLICATIONS
8. Standard threemode 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 abilityfactors study
13. Multivariate longitudinal data: Hospital study
14. Growth curves: Learningtoread study
15. Threemode correspondence analysis: Leiden electorate study
(Errata, 1989; available from author).

Kroonenberg, P.M. (1984a).
Centring threemode data: Views, problems, and queries. A
discussion with Harshman and Lundy. (WEP Reeks, WR 8454
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).
Threemode 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. 64103). New York: Praeger.
In this chapter, the threemode 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
example treating
data from attachment theory is used to illustrate some of the major
aspects and possibilities
of analyzing threemode data with the threemode principal component
model.

Kroonenberg, P.M. (1985).
Threemode principle components analysis of semantic differential data:
The case of a
triple personality. Applied Psychological Measurement, 9,
8394.
This paper shows how threemode 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 sumsof
squares
interpretations is explained and illustrated.

Kroonenberg, P.M. (1986).
The threemode world. (WEP Reeks, WR 8601LE), Leiden:
University
of Leiden, Department of Education.
This report is a rough attempt to show Who is Who in threemode land.
The emphasis is on
threemode factor and component analysis (i.e. threemode threeway
data), rather than on
individual differences multidimensional scaling (i.e. twomode threeway
data).

Kroonenberg, P.M. (1988b).
TUCKALS2. Threemode principal component analysis with extended
core matrix. In A. di Ciaccio & G. Bove (Eds.), Multiway
'88. Software Guide (pp. 93103). Roma:
Università di Roma "La Sapienza".
The main principles of threemode 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
pointed out.

Kroonenberg, P.M. (1988c).
TUCKALS3. Threemode principal component analysis. In A. di
Ciaccio & G. Bove (Eds.), Multiway '88. Software guide
(pp. 105114). Roma: Università di Roma "La
Sapienza".
The main principles of threemode PCA with full core array (or Tucker2model)
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 threeway
contingency tables. In R. Coppi & S. Bolasco (Eds.),
Multiway data analysis (pp. 169184). Amsterdam: Elsevier.
In this paper generalizations of the singular value decomposition are used
to analyze
interactions from threeway contingency tables. These decompositions are
primarily applied to
standardized residuals from various loglinear models to produce threeway
generalizations of
correspondence analysis.

Kroonenberg, P.M. (1989c).
The analysis of multiple tables in factorial ecology. III.
Threemode principal component analysis: "Analyse triadique
complète". Acta Oecologica. Oecologica Generalis,
10, 245256.
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 threemode factor (or principle
components) analysis. This
technique is applied to Thioulouse and Chessel's data on the water quality
of the
Méaudret. The compact and efficient data condensation of the method
is emphasized and
illustrated.

Kroonenberg, P.M. (1990b).
Threemode analysis by example. In Metodoloxía da
Investigacíon
Científica (pp. 105126). Santiago de Compostela, Spain:
Universidade. Servicio
de Publicacíons e Intercambio Científico.
In this paper threemode analysis, in particular, threemode principal
component analysis
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
threeway
techniques. For instance, the plant breeders' problem of evaluating
genotypes of soy beans in
different locations on various attributes for further selection has been
examined with
threemode 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 crosssectional data bases and repeated measures data can be
analysed fruitfully
with threeway methods.

Kroonenberg, P.M. (1992a).
PARAFAC in threeway land. [Comment on the article "Multilinear models:
applications in spectroscopy"]. Statistical Science, 7,
312314.
The purpose of this comment is to provide a somewhat wider background to
the PARAFAC model
discussed in Leurgans and Ross' paper on threeway methods in
spectroscopy. The literature on
the PARAFAC model is briefly sketched.

Kroonenberg, P.M. (1992b).
Threemode component models: A survey of the literature.
Statistica Applicata, 4, 619633.
This paper is a part of a larger review paper on threeway techniques. In
particular,
component models are reviewed, with special emphasis on PARAFAC and the
Tucker models.

Kroonenberg, P.M. (1993).
Threeway methods for multivariablemultioccasion
matrices. Paper presented at the 8th European Meeting of the
Psychometric Society, Barcelona, Spain.
A preliminary investigation is made into the usefulness of threemode PCA for
the analysis of multivariablemultioccasion (or multitraitmultimethod)
matrices. A brief comparison is made with stochastic threemode models (see
Bentler & Lee, 1978, 1979; Browne, 1984;
McDonald, 1984.)

Kroonenberg, P.M. (1994).
The Tuckals line: A suite of programs for threeway data analysis.
Computational Statistics
& Data Analysis, 18, 7396.
This paper describes two programs (tuckals2 en
tuckals3) with which threeway data can be
analysed. Both are based on generalisations of
standard (twoway) 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
drought conditions.

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
interface.

Kroonenberg, P.M. (1996b).
3WAYPACK User's manual: A package of threeway programs.
(Technical report+software), Leiden: University of Leiden, Department of
Education.
The collection of programs which constitute 3WAYPACK have been designed for the
analysis of threeway 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 userfriendly interface, INTERFACE3.

Kroonenberg, P.M. (1997a).
Recent developments in threeway data analysis: A showcase of methods and
examples.
In R. Klar & O. Opitz (Eds.), Classification and Knowledge
Organization (pp. 4462). Berlin: Springer.
In this paper a compact idiosyncratic overview will be provided of the areas
into which threeway 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
analyses.

Kroonenberg, P.M. (1998)
Studying the diffusion of threemode
analysis in chemistry: Design considerations. In A. Rizzi, M. Vichi
& H.H. Bock (Eds.), Advances in data science and
classification (pp. 603611). Berlin: Springer.
The paper presents an example of studying the introduction of
methodological innovation in a science, in particular that of
threemode (component) analysis into chemistry, especially in
such areas as chromatography, fluorescence, spectrometry, and
analytical chemistry in general.

Kroonenberg, P.M. (2001).
Threemode correspondence analysis: An illustrated exposé.
In: Actes des XXXIIIèmes Journée de Statistique
(pp. 101108). Nantes, France: 1418 May.
In this presentation, André Carlier's contributions to threemode
correspondence analysis will be presented. The special merit of his work
is that such tables are treated as genuine threeway arrays, rather than
as a matriced (or flattened) twoway matrix. The procedure will be
illustrated with an example describing the changes in play quality of
young children.

Kroonenberg, P.M., & Basford, K.E. (1989).
An investigation of multiattribute genotype response across
environments using threemode principal component analysis.
Euphytica, 44, 109123.
The usefulness of threemode principal component analysis to explore multi
attribute genotypeenvironment 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 ThreeMode Data Analysis. Chapter: ThreeMode 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 threemode 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).
Threeway cluster and component analysis of maize
variety trials.
Euphytica, 84, 3142.
Data from the Dutch variety list trials for maize
were analysed with threeway mixture method
clustering and threemode 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 threeway
mixture method clustering. British Journal of Psychology,
86, 397418.
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 threeway mixture method of clustering, a technique
especially designed for clustering threeway data (here: infants, variables and
episodes). The results are compared with the ABC 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 threeway
clustering procedure for data from the social and behavioural
sciences.

Kroonenberg, P.M., Basford, K.E., & Van Dam, M. (1992).
Threeway mixture method clustering. Annual Meeting
Classification Society of North America, 1213 June, 1992, East
Langsing.
This paper briefly explains the data  from 326 Dutch infants observed
in the
Strange Situation  and threeway mixture method clustering. It also
shortly dwells
upon the ordination technique to portray the cluster results. The major
objective
is to show threeway 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. 00177, 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).
TUCKALS2: Een
hoofdassenanalyse voor drieweggegevens. Methoden en Data
Nieuwsbrief (vd SWS vd VVS), 3 (3), 3053.
A condensed (Dutch) version of Kroonenberg & De Leeuw
(1977).

Kroonenberg, P.M. & De Leeuw, J. (1980).
Principal
component analysis of threemode data by means of alternating
least squares algorithms. Psychometrika, 45,
6997.
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.
(1973)

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, 6997.
One of the most serious problems in threedimensional quantitative structureactivity relationship (3DQSAR)
studies is selection of an alignment rule for molecular super position of the compounds in the data
set. In 3DQSAR analyses of structureactivity 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/alignmentdependent 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. 587597). 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 threemode principal component analysis of
Mann et al.'s data on crosscultural differences in respect for
others.
Journal of CrossCultural Psychology, 28, 463480.
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 doublecentred and both separately and
jointly analysed with threemode 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).
Threemode principal component analysis of multivariate
longitudinal organizational data. Sociological Methods &
Research, 14, 99136.
The exploratory role threemode 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,
105, 17.
Data on the physical growth of Japanese girls between 6 and 14 years old were
analysed to illustrate the application of threemode 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 threemode
analysis is discussed.

Kroonenberg, P.M. & Miyano, H. (1996c).
Threeway data analysis and its recent developments.
Japanese Psychological Review, 39, 386407.
Threeway data analysis methods are reviewed and explained from
the viewpoint of Tucker models. A simple description of a
threeway 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 threeway methods for the analysis of mortality trends
illustrated with worldwide female cancer mortality (19681985).
Statistical Methods in Medical Research, 11, 275292.
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 threeway methods applied
to a data set on female mortality rates for 17 tumour sites of 43
countries for the years 19681985. Multivariate techniques like biplots
and threemode principal component analysis within an overall threeway
analysisofvariance 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
threeway 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).
Threemode analysis of multimode covariance matrices.
British Journal of Mathematical & Statistical Psychology, 56, 305335.
Multimode covariance matrices, such as multitraitmultimethod matrices, contain
the covariances of subject scores on variables for different occasions or conditions.
This paper presents a comparison of threemode component analysis and threemode
factor analysis applied to such covariance matrices. The differences and similarities
between the nonstochastic 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 threemode factor analysis and its associated degrees of
freedom for the threemode component models. Furthermore, within the present
context a case is made for interpreting the core array as secondorder components.

Kroonenberg, P.M., & Snyder Jr, C.W. (1989).
Individual differences in assimilation resistance and affective
responses in problem solving. Multivariate Behavioral
Research, 24, 257284.
Data, comprising 6 judgment scales by 8 problem solving tasks by 32 thirteen
yearold boys, collected within the framework of Eckblad's (1981b) cognitive
theory of affect are analyzed with threemode principal component analysis. In
general, this study illustrates the effectiveness of the threemode principal
component analysis (TUCKALS) method for the assessment of the nomothetic
validity of a theoretical framework as it pertains to within and across person
variation.

Kroonenberg, P.M., & Ten Berge, J.M.F. (1987).
Crossvalidation of the WISCR factorial structure using
threemode principal components analysis and perfect congruence
analysis. Applied Psychological Measurement, 11, 195210.
By using threemode principal components analysis and perfect congruence
analysis in conjunction, the factorial structure of the 11 correlation matrices
of the Wechsler Intelligence Scale for ChildrenRevised 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 WISCR
factorial structure, the structures of other independently conducted studies,
and those of several translations of the WISCR 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 threemode 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 WISCR components in all studies without encountering rotation
problems.

Kroonenberg, P.M., & Ten Berge, J.M.F. (1989).
Threemode principal component analysis and perfect congruence
analysis for sets of covariance matrices. British Journal of
Mathematical and Statistical Psychology, 42, 6380.
In this paper threemode 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 threemode principal component
analysis close links between the two techniques exist. The methods are
illustrated with data pertaining to a theory of selfconcept.

Kroonenberg, P.M., Ten Berge, J.M.F., Brouwer, P.,
& Kiers, H.A.L. (1989).
GramSchmidt versus BauerRutishauser in alternating
leastsquares algorithms for threemode principal component
analysis. Computational Statistics Quarterly, 5,
8187.
The effect of replacing a BauerRutishauser step using an eigendecomposition by
a GramSchmidt orthogonalization step in an algorithm for threemode principal
component analysis was explored both theoretically and empirically. The results
showed that the latter procedure has a slight to moderate advantage over the
former one.

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, 117144.
Interactions of threeway factorial designs can be clarified by using
multiplicative decompositions, such as the singular value decomposition and
threemode principal component analysis. This procedure is illustrated for a
threeway analysisofvariance 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 BowlbyAinsworth 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 threemode 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, 281293.
Some methods that analyze threeway 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 threeway array, namely, the number of subjects,
the number of tests, and the number of treatments.

Kruskal, J.B. (1977).
Threeway arrays: Rank and uniqueness of trilinear
decompositions, with application to arithmetic complexity and
statistics. Linear Algebra and Its Applications, 18,
95138. (Corrections, 1711984; available from author or The Threemode
Company)
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,
x_{ijk} = a_{I}b_{j}c_{k}, whose
sum is X, and dim_{1}(X) to be the dimensionality of the
space of matrices generated by the 1slabs 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,
Sigma_{r}a_{ir}b_{jr}c_{kr}, to be
essentially unique. The results have applications to arithmetic complexity
theory and to statistical models used in threeway multidimensional
scaling.

Kruskal, J.B. (1981).
Multilinear models for data analysis. Behaviormetrika,
10, 120.
This paper is about structural models which are bilinear, trilinear, or
multilinear of higher order, such as the PARAFAC model, used to analyze twoway
arrays, threeway arrays or manyway 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, 75104.
This paper is about structural models which are bilinear, trilinear, or
multilinear
of higher order, used to analyze 2way arrays, 3way arrays or manyway
arrays
of higher order. Many of these models have geometrical content, which is
explained briefly. The bilinear models touched on include factor
analysis, principal
component analysis, and multidimensional scaling. Trilinear models
touched on
include INDSCAL and PARARAFAC. An application of the latter is presented.
Also, the new methods of preprocessing are described in an appendix,
because no
other description is available yet in the published literature.

Kruskal, J.B. (1983b).
Statement of some current results about threeway arrays.
Informal notes.
The concept of rank extends naturally to threedimensional 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
described.

Kruskal, J.B. (1985).
Rank of Nway arrays and the geometry of 2×2×2
arrays.
(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 manyway arrays of all orders. This extension is so useful and
natural, particularly for 3way arrays, that it has been introduced on several
separate occasions for different purposes. The smallest 3way 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 3way and Nway arrays.
In R. Coppi & S. Bolasco (Eds.), Multiway data analysis
(pp. 718). 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 3way array as a sum or linear
combination of outer product matrices underlies PARAFAC, and can be used to
define rank of 3way arrays. The many differences between 3way arrays and 2way
arrays with respect to decomposition and rank are discussed. For 2way 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 Nway arrays, which is closely connected to the
number of factors used in 3mode factor analysis.

Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1985).
Several mathematical relationships between PARAFACCANDECOMP and
3mode factor analysis. Paper presented at the Annual
Meeting of the Classification Society, St. John, Newfoundland,
Canada. [Published as Kruskal, Harshman &
Lundy, 1989].
Cf. Kruskal, Harshman, & Lundy, 1989.

Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1989).
How 3MFA data can cause degenerate PARAFAC solutions, among
other relationships. In R. Coppi & S. Bolasco (Eds.),
Multiway data analysis (pp. 115122). Amsterdam: Elsevier.
This paper discusses relationships among three models: (1) 3mode factor
analysis, (2) PARAFACCANDECOMP, 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).
Betweengroups comparison of principal components. Journal of
the American Statistical Association, 74,
703707.
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, 164168.
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).
Betweengroup analysis with heterogeneous covariance matrices: The
common principal component model. Journal of Classification,
7, 8198.
This paper proposes two methods for betweengroup analysis when
the common principal component model replaces the equal
dispersion matrix assumption. One method is by extension of the
twostage 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 betweenpopulations 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, 139144.
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 2dimensional excitation/
emission scans of samples at different pH, the 3dimensional 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
fluorescent species.

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, 5164.
Firstorder confirmatory factor analytic models have had
widespread use in the analysis of multitraitmultimethod (MTMM) data. In
contrast to the usual firstorder 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 firstorder 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
illdefined solutions frequently encountered with use of firstorder 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
research.

Kunert, J., & Qannari, E. M. (1999).
A simple alternative to generalized procrustes analysis: Application to sensory
profiling data.
Journal of Sensory Studies,14, 197208.
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,
195205.
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 39item social attitude
questionnaire once a year. In order to explore the factor structure, the Quasi
ThreeMode 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, 141158.
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 socalled 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.
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Centre for Child and
Family Studies 
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ThreeMode bibliography

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