Three-Mode Abstracts, Part R
With one can go to the index of
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|Ra | Rb |
Rc | Rd |
Re | Rf |
Rg | Rh |
Ri | Rj |
Rk | Rl |
Rm | Rn |
Ro | Rp |
Rq | Rr |
Rs | Rt |
Ru | Rv |
Rw | Rx |
Ry | Rz |
Rajko, R. (2001).
Calibration of chemical measurements. On quality of analytical information.
Magyar Kemial Folyoirat, 107, 45-59.
The author has been planning to publish a comprehensive review
on calibration for a long time. However, it seems almost
impossible task at all time. By all means it is necessary to
look back, because it turned out that there are several
emerging problems to investigate and to solve even in the
simplest linear univariate cases. Looking ahead is also needed,
because the difficult tensorial calibration models offer
fruitful developing. The calibration is placed in the flow of
the quality assurance of getting analytical information, and
the calibration models are examined in compliance with it, The
figures of merit are strongly important, they can serve as a
guide for instrument design and characterization. Several
questions have remained open for further scientific research.
It is hoped some future results are due to the contribution of
Ramaker, H.J., Van Sprang E.N.M., Gurden, S.P., Westerhuis J.A. &
Smilde A.K. (2002)
Improved monitoring of batch processes by incorporating external information
Journal of Process Control, 12, 569-576
In this paper an overview is given of statistical process monitoring with
the emphasis on batch processes and the possible steps to take for improving
this by incorporating external information. First, the general concept of
statistical process monitoring of batches is explained. This concept has
already been shown to be successful according to the number of references
to industrial applications. The performance of statistical process monitoring
of batch processes can be enhanced by incorporating external information.
Two types of external information can be distinguished: batch-run specific
and process specific information. Various examples of both types of external
information are given. Several ideas of how to incorporate the external
information in model development are discussed. The concept of incorporating
process specific information is highlighted by an example of a grey model.
This model is applied to a biochemical batch process that is spectroscopically
Ramsay, J.O., Munhall, K.G., Gracco, V.L., & Ostry, D.J.
Functional data analyses of lip motion.
Journal of the Acoustical Society of America, 99, 3718-
The vocal tract's motion during speech is a
complex patterning of the movement of many
different articulators according to many different
time functions. Understanding this myriad of
gestures is important to a number of different
disciplines including automatic speech
recognition, speech and language pathologies,
speech motor control, and experimental phonetics.
Central issues are the accurate description of the
shape of the vocal tract and determining how each
articulator contributes to this shape. A problem
facing all of these research areas is how to cope
with the multivariate data from speech production
experiments. In this paper techniques are
described that provide useful tools for describing
multivariate functional data such as the
measurement of speech movements. The choice of
data analysis procedures has been motivated by the
need to partition the articulator movement in
various ways: En effects separated from shape
effects. Partitioning of syllable effects, and the
splitting of variation within an articulator site
from variation from between sites. The techniques
of functional multivariate procedures such as
analysis of variance and principal components
analysis have their functional counterparts, and
these reveal in a way more suited to the data the
important sources of variation in lip motion.
Finally, it is found that the analyses of
acceleration were especially helpful in suggesting
possible mechanisms. The focus is on using these
speech production data to understand the basis
principles of coordination. However, it is
believed that the tools will have a more general
Ramsay, J.O., Ten Berge, J., & Styan, G.P.H. (1984).
Matrix correlation. Psychometrika, 49, 403-423.
A correlational measure for an n by p matrix X and
an n by
q matrix Y assesses their relation without specifying
either as a fixed target.
This paper discusses a number of useful measures of correlation, with
emphasis on measures
which are invariant with respect to rotations or changes in singular
values of either matrix.
The maximization of matrix correlation with respect to tranformations
XL and YM
is discussed where one or both transformations are constrained to be
attention is focussed on transformations which cause XL and
YM to be n
by s, where s may be any number between 1 and min(p,
efficient algorithm is described for maximizing the correlation between
YM where analytic solutions do not exist. A factor analytic
example is presented
illustrating the advantages of various coefficients and of varying the
number of columns of
the transformed matrices.
Ray, W.J., & Cole, H.W. (1985).
EEG alpha activity reflects attentional demands, and beta
activity reflects emotional and cognitive process. Science,
Two experiments were designed to examine the effects of attentional
demands on the
electroencephalogram during cognitive and emotional tasks. We found an
interaction of task
with hemisphere as ell as more overall parietal alpha for tasks not
requiring attention to the
environment, such as mental arithmetic, than for those requiring such
hemispheric activation for beta was found most strongly in the temporal
area for emotionally
positive or negative tasks and in the parietal areas for cognitive tasks.
These data were
analyzed by analysis of variance (day x attentional focus x emotional
valence x hemisphere)
and by a factor analytic technique (PARAFAC).
Ray, W.J., & Vasey, J. (1985).
PARAFAC analysis of psychophysiological data.
Psychophysiology, 22, 577.
In this presentation Ray briefly discusses the underlying model in comparison to
traditional factor analysis and illustrates the procedure with EEG data
collected in the context of a study examining the processing of positive and
negative emotion. The results of the PARAFAC analysis answer a variety of
questions for the psychophysiologists. First, the analysis points out any
outliers among the subjects. Second, the procedure shows how the EEG from the
various sites of the cortex contribute to the processing of particular tasks.
And third, the analysis can lend further support to the suggestion of the
experimenter that a priori groupings of tasks (e.g., positive and negative
emotional stimuli) are appropriately categorized.
Rayens, W.S., & Mitchell, B.C. (1997).
Two-factor degeneracies and a stabilization of
Chemometrics and Intelligent Laboratory Systems, 38, 173-181.
Mitchell and Burdick (B.C. Mitchell, D.S. Burdick,
An empirical comparison of resolution methods for
three-way arrays, Chemom. Intell. Lab. Syst. 20
(1993) 149-161; B.C. Mitchell, D.S. Burdick,
Slowly converging Parafac sequences: Swamps and
two-factor degeneracies, J. Chemom. 8 (1994)
155-168.) uncovered an intriguing correspondence
between the existence of Parafac swamps and the
presence of two-factor degeneracies (2FDs). This
observation, coupled with the recognition that
post-swamp resolutions were generally better than
pre-swamp resolutions allowed the user a method of
detecting when a swamp had likely been encountered
and, hence, when it was safe to assume that
Parafac had converged to a suitable resolution.
Still, this correspondence alone did not suggest
how one might reduce the number of Parafac
iterations required to move through a swamp. In
this paper, it is noted that serious 2FDs must
produce an identifiable ill-conditioning in the
least squares estimation step in Parafac.
Moreover, a serious 2FD is only one way this
ill-conditioning may occur and, hence, it is more
correct to say that swamps correspond to this
ill-conditioning in general, rather than to the
presence of 2FDs in particular. In an attempt to
reduce the number of iterations that parafac
spends in a swamp, a particular method of
stabilization is employed and results are
presented which suggest that the number iterations
can often be greatly reduced.
Rayner, J.C.W., Best, D.J., & Liddell, G.F. (1991).
Optimal testing in a three way ANOVA model. Communications in
Statistics - Simulation, 20, 411-425.
Likelihood ratio, score and Wald tests are considered for a three-way
random effects ANOVA
model. Competitor tests are compared using criteria such as small sample
relative efficiency, and convenient null distribution. The final choice
is between a new test
and two tests long used in practice.
Realo, A., Koido, K., Ceulemans, E., & Allik, J. (2002).
Three components of individualism. European Journal of Personality,
In this article, following an assumption that individualism and collectivism are separate
factors, we have further established that three central components of individualism can be
distinguished. In the first part of the article we examined whether the three proposed
components of individualism—autonomy, mature self-responsibility, and uniqueness—
can be distinguished from each other in one cultural context, Estonia. A new scale was
developed to measure the three aspects of individualism which demonstrated both the
reasonable internal-consistency reliability as well as convergent and divergent validity
with several other measures of individualism and collectivism and related constructs. In
the second part of the article we studied whether individualism generalizes across specific
contexts or domains of social relationships, namely, across relations with family and close
others; friends and peers; state and nation. The results of the three-mode principal
component analysis showed that the individualistic tendencies of the respondents did not
differ much while measured toward the three types of social relation.
Redfield, J. (1978).
TMFA: A FORTRAN program for three-mode factor-analysis and
individual-differences in multidimensional-scaling.
Educational and Psychological Measurement,
Description of a program to perform all three methods of
Tucker (1966) and also three-mode scaling (Tucker, 1972);
includes fitting of the additive constant in case of input of
similarities. Large number of options for input, analysis, and
Redfield, J. & Stone, A. (1979).
Individual view points of stressful life events. Journal of
Consulting and Clinical Psychology, 47, 147-
85 students rated 44 stressful life events on six 21 interval
bipolar scales. Analysis with T3. The events and scales
factors were graphically rotated to simple structure. The 3
student factors by varimax. The factors were scaled according
to Snyder & Wiggins (1970). Little numerical detail.
Validation with outside variables.
Reid, J.C. (1969).
A three-mode factor analysis of students' perceptions of a university.
The Journal of Experimental Education, 36, 93-96.
The structure of students' responses on a semantic differential
instrument to certain facets of their university was analyzes by
Tucker's three-mode factor analytical procedure. Factors were
identified and named for the R, P, Q and core matrices. Tucker's
procedure offers the investigator additional information in
interpreting such response data. The additional information
obtained from Tucker's procedure over classical two-mode
procedures, particularly the core box of interdependencies, was
helpful in interpreting the semantic differences data.
Reinikainen, S.-P., Laine, P., Minkkinen, P., & Paatero, P. (2001).
Factor analytical study on water quality in Lake Saimaa, Finland. Fresenius
Journal of Analytical Chemistry, 369, 727-732.
Multivariate data analysis methods (4-way Candecomp-PARAFAC: model solved
with Multilinear Engine (ME-1)) were used to interpret the data of over two
decades to study the changes in the water of Lake Saimaa in Finland. Earlier
studies have shown that it is difficult to extract the natural background
from the other sources of variation. By using the multilinear model three
interpretable factors representing natural and anthropogenic processes could
be extracted. The natural long-term variation, seasonal fluctuation and dilution
of discharges in the recipient area could be extracted into their own factors,
which could be easily visualized. The variation could be also presented with
estimated variation in the water quality parameters caused by each of these
natural or anthropogenic processes.
Reis, M.M., Gurden, S.P., Smilde, A.K., & Ferreira, M.M.C. (2000).
Calibration and detailed analysis of second-order flow injection analysis data
with rank overlap. Analytica Chimica Acta, 422, 21-36.
With the current popularity of second-order (or hyphenated) instruments, there
now exists a number of chemometric techniques for the so-called second-order
calibration problem, i.e. that of quantifying an analyte of interest in the
presence of one (or more) unknown interferent(s). Second-order instruments
produce data of varying complexity, one particular phenomenon sometimes
encountered being that of rank overlap (or rank deficiency), where the overall
rank of the data is not equal to the sum of the ranks of the contributing
species. The purpose of the present work is to evaluate the performance of two
second-order calibration methods, a least squares-based and an eigenvalue-based
solution, in terms of their quantitative ability and stability, as applied to
flow injection analysis (FIA) data which exhibits rank overlap. In the presence
of high collinearity in the data, the least squares methods is found to give a
more stable solution. Two-mode component analysis (TMCA) is used to investigate
the reasons for this difference in terms of the chemical properties of the
species analysed. The success of second-order calibration of this data is found
to depend strongly on the collinearity between the acidic and basic time
profiles and the reproducibility of the pH gradient in the FIA channel, both of
which are shown to be related to the pKa values of the
Reis, M. M., Biloti, D. N., Ferreira, M. M. C., Pessine, F. B. T., & Teixeira, G. M. (2001).
PARAFAC for spectral curve resolution: A case study using total lumniscence in human dental tartar.
Applied Spectroscopy, 5, 847-851.
Chromophore identification in biological samples often requires the
physical separation of the compounds, which can be difficult. Although there are
several advantages to hyphenated spectroscopic techniques for identification of
substances, complex mixtures of chromophores presenting overlapped spectra cannot
be identified directly through this method. This work presents an application of
chemometrics to compound identification in biological samples by a spectroscopic
hyphenated technique using a curve resolution method. The PARAllel FACtor analysis
model (PARAFAC), which has no rotational indeterminacy, was used for curve resolution
of excitation-emission spectra of human dental tartars. PARAFAC was applied under
constraints (i.e., unimodality and non-negativity) and evaluated with a validation
procedure. The resolved profiles are porphyrinic-like spectra presenting excitation
band maxima at 407, 416, and 431 nm in the Soret band region (390–440 nm) of these
Reis, M. M., & Ferreira, M. M. C. (1999).
Curve resolution of simulated and total luminescence spectra by Generalized Rank Annihilation Method (GRAM).
Quimica NOVA, 22, 11-17.
A multivariate curve resolution method, "generalized rank annihilation method (gram)", is discussed
and tested with simulated and experimental data. The analysis of simulated data provides general guidelines
concerning the condition for uniqueness of a solution for a given problem. The second-order emission-excitation
spectra of human and animal dental calculus deposits were used as an experimental data to estimate the performance
of the above method. Three porphyrinic spectral profiles, for both human and cat, were obtained by the use of gram.
Reis, M. M., & Ferreira, M. M. C. (2002b).
PARAFAC with splines: a case study. Journal of Chemometrics, 16, 444-450.
The PARAFAC model has been used in several applications in chemistry, e.g. for overlapped
spectra resolution and second-order calibration. In general, the PARAFAC method uses a vector space approach
by considering the matrices resulting from the decomposition as a collection of vectors. This paper presents
a PARAFAC application where the factors resulting from the decomposition are considered as functions. The
functional objects used for this are spline functions. The methodology used performs the Spline-PARAFAC
decomposition based on the Bro-Sidiropoulos approach for the unimodality constraint. One of the advantages
of using splines is the possibility of achieving a controlled degree of smoothing on the decomposed components.
The amount of smoothing applied on the components in the presented methodology is controlled by a penalty
parameter or by the number of basis functions. Thus Spline-PARAFAC requires the calculation of the parameter
and the number of basis functions, which were determined in this work by using ordinary cross-validation (OCV).
Spline-PARAFAC was applied to a carbon monoxide data set comprising concentrations measured every hour during
the years 1997 and 1999 in the city of São Paulo, Brazil. Each data set was arranged in a three-way array of
dimension (24 hours × 5 days × 52 weeks). Spline-PARAFAC showed a good performance, producing smoothed
profiles describing the daily variations in emitted gas and the seasonal effects during the year.
Reis, M. M., Ferreira, M. M. C. & Sarmento, S. B. S. (2002c).
A multi-way analysis of starch cassava properties. Chemometrics and Intelligent Laboratory Systems,
The original methods proposed by Ledyard R. Tucker during the 1960s present the rotational freedom problem, making the
interpretation of their results rather difficult to be carried out. Aiming to make the multi-way data analysis more acceptable, this
work suggests a methodology for extracting meaningful information from the data set. This methodology is based on the
decomposition of data set in three-way blocks by using Tucker models. With the aim of keeping in one block similar
information about the data properties, a decomposition based on a constrained Tucker model was used, where the core array has
some of its elements fixed to zero. This methodology is successfully applied to a data set formed by physical and
physicochemical properties of starches of four cassava cultivars, harvested at different ages during the period usually taken for
harvest of industrial uses.
Renaud, J.P., Davydov, D.R., Heirwegh, K.P.M., Mansuy, D.,
Hoa, G.H.B. (1996).
Thermodynamic studies of substrate binding and
spin transitions in human cytochrome p-450 3a4
expressed in yeast microsomes.
Biochemical Journal, 319, 675-681.**
An approach to the quantitative spectral analysis
of substrate binding and inactivation of
cytochrome p-450 in microsomes is described. The
method is based on the application of the
principal component analysis technique on the
soret-region spectra measured at different
temperatures at various concentrations of
substrate. This approach allowed us to study the
thermodynamic parameters of substrate binding and
spin transitions in human cytochrome p-450 3a4
expressed in yeast (saccharomyces cerevisiae)
microsomes. These parameters are discussed in
comparison with the values reported earlier by
ristau et al. [(1979) Acta Biol. Med. Ger. 38,
177-185] for rabbit liver cytochrome p-450 2b4 in
solution with benzphetamine as a substrate. Our
analysis shows the substrate-free states of 2b4
and 3a4 to be very similar. However, substrate
binding seems to perturb haem-protein interactions
in 3a4 in contrast with 2b4, where the effect of
substrate binding on the thermodynamic parameters
of spin transitions was insignificant. The
implication of the results for the mechanism of
substrate-induced spin shift is discussed.
Reyment, R.A. (1997).
Multiple group principal component analysis. Mathematical Geology,
Common Principal Component Analysis is a generalization of standard principal
components to several groups under the rigid mathematical
assumption of equality of all latent vectors across groups (i.e., principal
component directions), whereas the latent roots are allowed to vary
between groups (differing inflations of dispersion ellipsoids). In practice,
data that fulfill these strict requirements are relatively rare. Examples
from palaeontology are used to illustrate the principles. Compositional data can
be made to fit the Common Principal component (CPC) model by
the appropriate logratio covariance matrix.
Reyment, R.A., & Naidin, D.P. (1962).
Biometric study of actinocamax verus s.1. from the upper
cretaceous of the Russian platform. Contributions in
Geology. Acta Universitatis
Stockholmiensis, 9, 147-206.
This paper studies belemnite species from 9 localities. Primarily interesting
because it contains a set of correlation matrices which were analysed later with
three-mode methods (see Reyment,
Richards, E., Bessant, C., & Saini, S. (2003).
Multivariate data analysis in electroanalytical chemistry.
Electroanalysis, 14, 1533-1542.
Data analysis is becoming an increasingly important aspect of
electroanalytical chemistry, as voltammetric techniques and electrode arrays become ever
more popular as diagnostic tools. Modern data analysis techniques promise to help us make
full use of the large amounts of data collected, allowing electroanalytical chemists to
get more out of their existing instruments, and paving the way for new measurement
approaches. Ibis article provides an overview of the most widely used multivariate
techniques in electroanalysis, citing specific examples of how they have been applied,
and looking at their relative merits. As in other areas of analytical science, no single
technique is applicable to all applications and the running of controls and appreciation
of the applications and limitations of each technique is essential.
Rincon, F., Johnson, B., Crossa, J., & Taba, S. (1997).
Identifying subsets of maize accessions by three-mode principal component analysis.
Crop Science, 37, 1936-1942.
Genebank accessions are a potential source of genetic
variability for maize (Zea mays L.) breeding programs. Identification of
useful individual entries is commonly based on the expression of one or
more attributes in different sites or environments. This study was
motivated by the need to identify a subset of Caribbean accessions for
introgression into elite germplasm of the western Corn Belt. Thus the
objective,vas to identify potentially useful Caribbean accessions, based
on (i) simultaneous consideration of six agronomic attributes deemed
economically important for the western Corn Belt, and (ii) response
patterns observed across four sharply contrasting environments. Both (i)
and (ii) were addressed by means of three-mode principal component
analysis (PCA) of data on six agronomic and morphological attributes for
184 Caribbean maize accessions evaluated at four environments. Three-way
data were analyzed by three-mode PCA, and based on (i) and (ii), two
joint plots were generated. From the PCA and joint plots, two subsets of
accessions were identified. First, a subset of 14 accessions having good
yield, intermediate plant height, and average days to anthesis was
identified. Secondly, a subset of 10 accessions having average
performance over all environments was identified. Two accessions were
common to both subsets. Jointly, the two approaches produced a combined
subset of 22 accessions, representing 12% of the total evaluated, and
including representatives of 11 maize races. Three-mode PCA integrated
information on accessions, attributes, and environments, and provided a
means of simultaneously visualizing these three types of information.
Three-mode PCA can complement standard methodologies used by plant
breeders for identification of potentially useful accessions in
Rikken, F., Kiers, H.A.L., & Vos, R. (1995).
Mapping the dynamics of adverse drug reactions in subsequent
time periods using INDSCAL. Scientometrics, 33, 367-380.
In this study we have focused on the problem of mapping the dynamics of
co-word-matrices from subsequent time periods. Methods for mapping dynamics are
important for following trends in research. We have explored the possibilities of
a three way multidimensional scaling method, INDSCAL. We are especially interested
to find relations between adverse drug reactions and other words in co-word-matrices
from a medical field. Second we want to explore whether the relations between
adverse drug reactions and other words have changed in subsequent time periods.
The results show that INDSCAL can be a useful tool for mapping dynamics.
Riu, J., & Bro, R. (2003).
Jack-knife technique for outlier detection and estimation of standard errors in
PARAFAC models Chemometrics and Intelligent Laboratory Systems, 65, 35-49.
In the last years, multi-way analysis has become increasingly important because
it has proved to be a valuable tool, e.g. in interpreting data provided by
instrumental methods that describe the multivariate and complex reality of a given
problem. Parallel factor analysis (PARAFAC) is one of the most widely used multi-way
models. Despite its usefulness in many applications, up to date there is no
available tool in the literature to estimate the standard errors associated with
the parameter estimates. In this study, we apply the so-called jack-knife technique
to PARAFAC in order to find the associated standard errors to the parameter estimates
from the PARAFAC model. The jack-knife technique is also shown to be useful for
detecting outliers. An example of fluorescence data (emission/excitation landscapes)
is used to show the applicability of the method. (C) 2002 Elsevier Science B.V.
All rights reserved.
Rizzi, A. (1988).
On the average matrix. In H.H. Bock (Ed.), Classification and
related methods of data analysis (pp. 533-540).
This paper proposes to construct a mean matrix of a set of matrices such that
the standard deviations for each averaged variable is equal to the mean standard
Rizzi, A. (1989a).
On the principal component analysis of three-way data matrices.
In Y. Dodge (Ed.), Statistical Data Analysis and Inference, 501-
This paper is concerned with the Principal Component Analysis
of three-way data matrices. Principal matrices are a linear
combination of K matrices observed on K different occasions or
sources of data; they have maximum Euclidean norm and they are
incorrelated. After some remarks on centering three-way data
matrices, the author introduces, also, some new definitions for
the standardization of multiple matrices. Applications are found
at the end of the paper.
Rizzi, A. (1989b).
On the synthesis of three way data matrices. In R. Coppi & S.
Bolasco (Eds.), Multiway data analysis (pp. 143-154).
In this paper a new type of matrix is proposed which has the property that the
deviance of the variables is equal to the mean deviance. Also several principal
matrices are proposed and their analysis is discussed.
Rizzi, A. (1989c).
Clustering per le matrici a tre vie [Clustering for three-way
matrices]. Statistica, 49, 195-208.
Cluster analysis for three-way and three-mode data matrices can be done in
various ways, in
particular any of the three modes may be clustered, or the clustering can
involve more than
one mode. In this paper, after reviewing the INDCLUS and ADCLUS
algorithms, two more recent
proposals by Bellacicco & Ronzoni (1988) and Chiodi (1989) are
illustrated. In addition a
new algorithm, PRINCLUS, based on principal matrices is
Rizzi, A., & Vichi, M. (1992).
Relations between sets of variates of a three-way data set.
Statistica Applicata, 4, 635-651.
In this paper we analyze two types of relations between sets of
variates: the correlation and the connection (or dependence).
Their measures can be derived from a general index that is
examined. Furthermore, we discuss new results on Principal
Matrices Analysis (PMA) and Factorial Matrices Analysis (FAMA),
two techniques based on the relations between sets of variates of
a three-way data matrix due to the sharing of common sets of
Rizzi, A., & Vichi, M. (1995a).
Representation, synthesis, variability and data preprocessing of
a three-way data set. Computational Statistics & Data
Analysis, 19, 203-222.
In the first section of this paper the structures (vectors and matrices)
on which three-way
data set X can be organized is described, as well as the
information that can
be pointed out when using these structures. Many three-way analyses are
based on pooled
representations of X, that are systematically studied. The
by a three-way data set can be synthesized according to the structures
utilized to represent
X. In the second section one-way, two-way and three-way
X are defined. Also the variability of a three-way data
set is evaluated, in
section three, according to three different levels: one-way or fiber
variability, two-way or
slab variability and three-way variability. The syntheses and the
variability indices of
X can be used for data preprocessing of X,
which is here
discussed in section four. Furthermore, section five discusses the
Analysis on the base of three-way variability indices.
Robert, P., & Escoufier, Y. (1976).
A unifying tool for linear multivariate statistical methods: The
RV-coefficient. Applied Statistics, 25,
Consider two data matrices on the same sample of n individuals,
× n), Y(q × n). From these
representations of the sample are obtained as two configurations of
n points, in
domain Rp and domain
Rq. It is shown that
the RV-coefficient (Escouffier, 1970, 1973) can be used as a
measure of similarity of
the two configurations, taking into account the possibly distinct metrics
to be used on them
to measure the distances between points. The purpose of this paper is to
show that most
classical methods of linear multivariate statistical analysis can be
interpreted as the
search for optimal linear transformations or, equivalently, the search
for optimal metrics to
apply on two data matrices on the same sample; the optimality is defined
in terms of the
similarity of the corresponding configurations of points, which, in turn,
calls for the
maximization of the associated RV-coefficient. The methods studied
components, principal components of instrumental variables, multivariate
canonical variables, discriminant analysis; they are differentiated by
relationships existing between the two data matrices involved and by
under which the maximum of RV is to be obtained. It is also shown
RV-coefficient can be used as a measure of goodness of a solution
to the problem of
Robert, P., Bertrand, D., & Sire, A. (1992).
Identification of chemical-constituents by multivariate near-infrared spectral imaging.
Analytical Chemistry, 64, 664-667.
A near-infrared imaging spectroscopic system was tested to identify three main components of wheat (bran,
gluten, starch). The system described below permitted the recording of images between 900 and 1900 nm by steps of 50 nm. A
direct examination of images showed that starch was correctly identified at 1550 nm. However, such a direct study of images
was not sufficient to characterize all the constituents. The images, therefore, were linearly combined by applying discriminant
analyses. The more revelant wavelengths (950, 1450, 1500 nm) were determined using a stepwise discriminant analysis, and a
mapping of the chemical constituents was obtained by applying a canonical discriminant analysis. In the segmented image, the
percentages of well-classified pixels were 92 for bran, 95 for gluten and 99 for starch.
Rocci, R. (1992a).
Scalar product and synthesis of s-matrices. Statistica
Applicata, 4, 693-699.
In this paper an alternative view is given of the Euclidean scalar
product between symmetric
positive semidefinite matrices, characterizing a matrix on the grounds of
decomposition. Following this approach we reconsider the "compromise
matrix" and "mean
matrix" methods tacking into account the rank of the "compromise" or
Rocci, R., & Ten Berge, J.M.F. (1994).
A simplification of a result by Zellini on the maximal rank of symmetric three-way
arrays. Psychometrika, 59, 377-380.
Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix
of order n × n as a linear combination of 1/2n(n + 1)
fixed rank one matrices, thus constructing an explicit tensor basis for the set of
symmetric n × n matrices. Zellini's decomposition is based on
properties of persymmetric matrices. In this paper, a simplified tensor basis is
given, by showing that a symmetric matrix can also be decomposed in terms of
1/2n(n + 1) fixed binary matrices of rank one. The decomposition
implies that an n × n × p array consisting of p
symmetric n × n slabs has maximal rank 1/2n(n + 1).
Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of
such an array will yield a perfect fit in 1/2n(n + 1) dimensions. When
the fitting only pertains to the off-diagonal elements of the symmetric matrices,
as is the case in a version of PARAFAC where communalities are involved, the maximal
number of dimensions can be further reduced to 1/2n(n - 1). However,
when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis
result does not apply. In fact, it is shown that in this case the number of dimensions
needed can be as large as p, the number of matrices analyzed.
Rocci, R., & Ten Berge, J. M. F. (2002).
Transforming three-way arrays to maximal simplicity Psychometrika, 67, 351-365.
Transforming the core array in Tucker three-way component analysis to simplicity
is an intriguing way of revealing structures in between standard Tucker three-way
PCA, where the core array is unconstrained, and CANDECOMP/PARAFAC, where the core
array has a generalized diagonal form. For certain classes of arrays, transformations
to simplicity, that is, transformations that produce a large number of zeros, can
be obtained explicitly by solving sets of linear equations. The present paper extends
these results. First, a method is offered to simplify J x J x 2 arrays. Next, it is
shown that the transformation that simplifies an I x J x K array can be used to also
simplify the (complementary) arrays of order (J K - I) x J x K, of order I x (I K - J)
x K and of order I x J x (I J - K). Finally, the question of what constitutes the
maximal simplicity for arrays (the maximal number of zero elements) will be considered.
It is shown that cases of extreme simplicity, considered in the past, are, in fact,
cases of maximal simplicity.
Rodriguez-Cuesta, M. J., Boque, R., Rius, F. X., Zamora, D. P., Galera, M. M., & Frenich, A. G. (2003).
Determination of carbendazim, fuberidazole and thiabendazole by
three-dimensional excitation-emission matrix fluorescence and parallel
Analytica Chimica Acta, 491, 47-56.
We simultaneously determined carbendazim, fuberidazole and
thiabendazole by excitation-emission matrix (EEM) fluorescence in combination
with parallel factor analysis (PARAFAC). Three-way deconvolution provided the
pure analyte spectra from which we estimated the selectivity and sensitivity of
the pesticides, and the relative concentration in the mixtures from which we
established a linear calibration. Special attention was given calculating such
figures of merit as precision, sensitivity and limit of detection (LOD), derived
from the univariate calibration curve. The method, which had a relative precision
of around 2-3% for the three pesticides, provided limits of detection of 20 ng ml(-1)
for carbendazim, 4.7 ng ml(-1) for thiabendazole and 0.15 ng ml(-1) for fuberidazole.
The accuracy of the method, evaluated through the root mean square error of prediction
(RMSEP), was 27.5, 1.4, and 0.03 ng ml(-1), respectively, for each of the pesticides.
Röhmel, J., Streitberg, B., & Herrmann, W. (1983).
The COMSTAT algorithm for multimodal factor analysis: An
improvement of Tucker's three-mode factor analysis method.
Neuropsychobiology, 10, 157-163.
Three-mode factor analysis, as developed by Tucker (1966), is a
method for nonredundant representation of data arrays with three
subscripts. Tucker, however, did not succeed in obtaining a
least-squares solution for his model. In this paper a necessary
condition is derived for a least-squares solution and an
algorithm is constructed which improves any given initial
solution in the least-squares sense. Shown is that this algorithm
converges to a representation satisfying the necessary conditions
for a least-squares solution. The method is in no way restricted
to three dimensions, but can be applied to any multimodel data
Rösler, F. (1972).
Dimensionen der Aktivitäten und deren Beziehungen zu
den Persönlichkeitsfaktoren "Extraversion/Introversion" und
"Neurotizismus" sensu Eysenk. Unpublished master thesis,
University of Hamburg, Hamburg, FRG.
(See Rösler, 1975.)
Rösler, F. (1975).
Die Abhängigkeit des Elektroenzephalogramms von den
Persönlichkeitsdimensionen E und N sensu Eysenk und
unterschiedlich aktivierenden Situationen. Zeitschrift für
experimentelle und angewandte Psychologie,
As part of a larger study a reduction of frequency spectra was
obtained via T3 on the data of 32 subjects, 18 measuring
periods, and 40 frequencies. However, only principal
components was performed on the frequencies mode. The factors
were scaled as suggested by Bartussek (1973).
Rösler, F. (1979).
Identifying interindividual judgment
INDSCAL or three-mode factor analysis. Multivariate
Behavioral Research, 14, 145-167.
Seventy subjects judged 9 German key-politicians on 22 bipolar
7-point attribute scales. The proximity matrices from the
computed profile similarities were analyzed with INDSCAL. T3
was applied to the original data. The results suggested that
T3 was more appropriate to describe the interindividual
judgement differences when correlated with external
Rösler, F., Jesse, J., Manzey, D., & Grau, U. (1982).
Ist das LM-Gitter nur ein LM-Test? Eine dreimodale Faktorenanalyse des
LM-Gitters für Kinder (Schmalt). (Is the LM-Gitter only an LM
trimodal factor analysis of the LM-Gitter für Kinder (Schmalt).)
Diagnostica, 28, 131-145.
Three-mode factor analysis was used to check Need Achievement
Scale for Children for systematic situation-specific effects. Data
of 104 boys and 94 girls were analyzed. Factorization of the
statement mode resulted in a factor structure that separated the 3
subscales of the test in the same manner as reported by Schmalt.
Factorization of the situation mode reveals a Woodrow structure
only, thus pointing to time-correlated changes in response
behavior and not to differences related to the motivational
quality of the situations.
Ross, R.T., Lee, C.-H., Davis, C.M., Ezzeddine, B.M., Fayyad,
E.A., & Leurgans, S.E. (1991).
Resolution of the fluorescence spectra of plant
pigment-complexes using trilinear models.
Biochimica et Biophysica Acta, 1056, 317-320.
The intensity of fluorescence from a pigment is
separately linear in functions of excitation
wavelength, emission wavelength, and any chemical
treatment which alters overall fluorescence yield.
This multiple linearity permits the use of an
extension of principal components analysis to
resolve overlapping spectra without the use of any
additional information. The method is used to
resolve the spectra of pigment complexes in pea
thylakoids, using the concentration of Mg2+ as the
chemical treatment variable. Two components could
be resolved accurately. One has little or no
dependence on Mg2+; the other, with an excitation
spectrum resembling LHC II, has a dependence on
Mg2+ which follows the hill equation with a
binding constant of 0.4-0.6 mM and a hill
coefficient of 2.4-3.1.
Ross, R.T., & Leurgans, S.E. (1995).
Component resolution using multilinear models. In K. Sauer (Ed.),
Biochemical spectroscopy (pp. 679-700).
Diego: Academic Press. [Methods in Enzymology, 246].
In many circumstances, the spectroscopic properties of a biological specimen are
a composite of the properties of several different chromophores within the
specimen, and the investigator would like to know the properties of the
individual chromophores. Physical separation of the chromophores is often
difficult. Thus, one looks to mathematical methods for separation of the
contributions of the different chromophores. This chapter focuses on the
application of multilinear models. A major advantage of this class of methods is
that component resolution may often be achieved with no prior information about
the properties of the components. Bilinear models can be applied to many kinds
of spectroscopy, and they have been widely used. In this chapter the focus is on
the application of trilinear models in fluorescence spectroscopy. However, these
trilinear models are also applicable to other kinds of excited-state
spectroscopy, such as transient absorption spectroscopy. Along the way, bilinear
and other models, including global analysis, are also discussed.
Rowe, H.A.H. (1979).
Three-mode factor analysis: Problems of
interpretation and possible solutions. Australian
Psychologist, 14, 222-223 (abstract of
paper presented at the 14th Annual Conference of the Australian
Psychological Society, University of Tasmania, 30. August,
A summary of applications with T3 is presented. An example (18
problem solving strategies on 12 tasks of varying content and
difficulty by 89 subjects) is discussed on a conceptual level; no
numerical details are given. Special attention is given to
meaningful interpretations of the core matrix in relation to
different types of standardizations of the raw data.
Reification with external variables.
Ruch, W. (1980).
Gemeinsame Strukturen in Weizbeurteilung und
Persönlichkeit. Versuch einer empirischen Integration des
Witzbeurteilung in die differentielle Psychologie.
Unpublished doctoral thesis, University of Graz, Austria.
110 subjects rated 48 jokes on 5 seven-point scales. T3 using
the scaling of Bartussek (1973). The subject factors were in-
terpreted using external variables. Standardization over all
jokes-scales combinations. Varimax rotation for jokes, promax
for scales. Detailed analysis of subjects and core matrix
using 'ideal subjects' and the extended core matrix.
Ruch, W. (1981).
Witzbeurteilung und Persönlichkeit: Ein trimodale analyse.
Zeitschrift für Differentielle und Diagnostische
Psychologie, 2, 253-273.
110 subjects rated 48 jokes on 5 seven-point scales. T3 using
the scaling of Bartussek (1973). The subject factors were in-
terpreted using external variables. Standardization over all
jokes-scales combinations. Varimax rotation for jokes, promax
for scales. Detailed analysis of subjects and core matrix
using 'ideal subjects' and the extended core matrix.
Ruckebusch, C., Duponchel, L., Sombret, B., Huvenne, J. P., & Saurina, J. (2003).
Time-resolved step-scan FT-IR spectroscopy: Focus on multivariate curve resolution.
Journal of Chemical Information and Computer Sciences, 43, 1966-1973.
The present paper describes the application of step-scan FT-IR spectroscopy
in combination with chemometric analysis of the spectral data for the study of the photocycle
of bacteriorhodopsin. The focus is on the performance of this instrumentation for time-resolved
experiments. Three-dimensional data-spectra recorded over time-are studied using various factor
analysis techniques, e.g., singular values decomposition, evolving factor analysis, and multivariate
curve resolution based on alternating least squares. Transient intermediates formed in the time
domain ranging from 1 mus to 6.6 ms are clearly detected through reliable pure time evolving
profiles. At the same time, pure difference absorbance spectra are provided. As a result, valuable
information about transitions and dynamics of the protein can be extracted. We conclude first that
step-scan FT-IR spectroscopy is a useful technique for the direct study of difficult photochemical
systems. Second, and this is the essential motivation of this paper, chemometrics provide a step
forward in the description of the photointermediates.
Ruckebusch, C., Duponchel, L., Huvenne, J. P., & Saurina, J. (2004).
Multivariate curve resolution of step-scan FTIR spectral data.
Vibrational Spectroscopy, 35, 21-26.
Step-scan FTIR is a time-resolved spectroscopic technique. It can
successfully probe transient photochemical systems through two-way data sets: spectra are
recorded as the system is evolving with time.
This work focuses on chemometrics for time-resolved data sets and in particular on the
potential of the multivariate curve resolution (MCR) based on alternating least squares.
The usefulness of this approach is demonstrated in a case study, which involves several
intermediate states with time constant ranging from the microsecond to the millisecond.
The chemometrics methodology is detailed in two points.
MCR relies on the bilinear decomposition of an evolving multi-component data set into
the product of two matrices related respectively to the time and to the spectral domains
of the extracted components.
The first aspect concerns the determination of the number of components observable in
the spectral data set and the calculation of initial estimates of either the time or
spectral profiles. Principal component analysis based methods such as evolving factor
analysis are performed and finally allow a first insight into the contributions of the
four intermediates extracted.
The second point concerns the alternating least squares optimization under natural
constraints to recover pure intensity profiles and pure difference spectra profiles.
The characterization of each photo-intermediate unaccessible individually except in
non-physiological conditions is thus assessed without any assumption about the kinetic
of the system.
Russell, D. (1982).
The Causal Dimension Scale: A measure of
how individuals perceive causes. Journal of Personality and Social
Psychology, 42, 1137-1145.
The studies reported describe the development of the Causal
Dimension Scale, a measure designed to assess how the attributor
perceives the causes he or she has stated for an event. A
three-mode factor analysis confirmed the three-dimensional
structure of the scale.
Rychlak, J.F., Flynn, E.J. & Burger, G. (1979).
Affection and evaluation as logical processes of meaningfulness
independent of associative frequency. Journal of General
Psychology, 100, 143-157.
Forty-three undergraduates rated 25 CVC-trigrams and 25 words
and paralogs according to 7 different instructions. The
trigrams and the words-paralogs were analysed separately with
T3. Only the instructions were interpreted in both cases, as
only one factor was present in the verbal materials and the
subject matrix suggested the presence of only one factor.
Because of the decision to retain only one factor for subjects
and word materials, it was decided not to carry out a 'core
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Department of Educational
The Three-Mode Company |
Faculty of Social and Behavioural Sciences, Leiden University
The Three-Mode Company, Leiden, The Netherlands
E-mail: kroonenb at fsw.leidenuniv.nl
First version : 12/02/1997;