Three-Mode Abstracts, Part A
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Abdollahi, H., & Nazari, F. (2003).
Rank annihilation factor analysis for spectrophotometric study of complex
Analytica Chimica Acta, 486, 109-123.
The use rank annihilation factor analysis (RAFA) for spectrophotometric studies of
complex formation equilibria are proposed. One-step complex formation and two successive and
mononuclear complex formation systems studied successfully by proposed methods. When the complex
stability constant acts as an optimizing object, and simply combined with the pure spectrum of ligand,
the rank of original data matrix can be reduced by one by annihilating the information of the ligand
from the original data matrix. The residual standard deviation (R.S.D.) of the residual matrix after
bilinearization of the background matrix is regarded as the evaluation function. The performance of
the method has been evaluated by using synthetic data. For two-step successive complex formation
systems, the effects of noise level and equilibrium constants K-1 and K-2 on output of algorithm are
investigated. The applicability of method for resolving the two-step successive complex formation
systems with full spectral overlapping of two complex species also is shown. Spectrophotometric
studies of murexide-calcium, dithiazone-nickel and methyl thymol blue (MTB)-copper are used as
experimental model systems with different complexation stoichiometries and spectral overlapping of
Abel, R.B., Leurgans, S.E., & Ross, R.T. (1992).
Multilinear models: Experimental design in spectrofluoroscopy. Technical
Report No. 470. Department of Statistics, Ohio State University.
Multilinear models, particularly PARAFAC models, are useful for spectrofluoroscopic
data analysis. Experimental design techniques for nonlinear models can be applied
to PARAFAC models to guide spectrofluoroscopic data collection. This report concentrates
on the design criterion of D-optimality, the maximization of the determinant of the
model's information matrix. The information matrix for a PARAFAC model has two
distinguishing features. First, the information matrix is a function of inner and
outer products of the spectra. Second, although the dimensions of a PARAFAC model's
information matrix can be large, the determinant of the information matrix is a
simple function of determinants of smaller matrices. Searches for D-optimal designs
using this function were dramatically faster than searches using the entire information
matrix. Spectrofluoroscopic data collection often results in missing observations.
The optimality properties of some strategies for systematically avoiding observations
are explored. Another design criterion examined is that of minimizing the asymptotic
variance of the ratios of estimated absorption or emission intensities. This criterion
often results in different optimal designs than the criterion of D-optimality, but
D-optimal designs generally provide nearly minimal variance of ratio designs.
Aberg, P. (2002).
Electronic biopsies applications and data analysis.
Doctoral thesis, Karolinska Institutet, Stockholm.
Background/aims: This thesis presents two studies using electrical skin impedance, and an investigation
concerning how information can be extracted from multivariate impedance spectra. The two investigations
are sin property variations within volar forearms, and assessment of impedance of skin cancer and other
lesions. Various numerical methods are described, specially multivariate techniques.
Methods: Skin impedance spectra and trans epidermal water loss of volar forearms were measured on 8 sites
covering approcimately 4x4 cm2 the volar forearms of 27 healthy volunteers. Skin property variations were
investigated using 3-way ANOVA and parallel factor analysis. Impedance spectra on lesions were measured at
258 benign pigmented cellular nevi (BEN), 34 basal cell carcinoma (BCC), 17 dermatofibroma (DER), 35 dysplastic
nevi (DYS), and 26 seborrheic keratoses (SEB). Reference skin was measured ipsy-laterally to the lesions.
Differences between lesions and references were analysed using non-parametric statistics.
Results: 3-way ANOVA showed significant differences between inner and outer sides of forearms for impedance
(P<0.001) and trans epidermal water loss (P<0.01). Additional significant variations (P<0.01) between left and
right arm were shown using parallel factor analysis on the impedance spectra. Significant differences between
lesions and references were found for BEN (P<0.001), BCC (P<0,001), DYS (P<0.01), and SEB (P<0.01).
Conclusions: It is crucial to design skin studies carefully to avoid the baseline variations of skin to interfere
with the studied phenomenon, and impedance methods might, after technical developments, be used as a diagnostic
decision tool for skin lesions. However, the information distribution in the ingerent in the impedance spectra is
such that, for subtle diagnostic applications, appriopriate number-crunching techniques are necesssary.
Aberg, P., Geladi, P., Nicander, I. & Ollmar, S. (2002).
Variation of skin properties within human forearms demonstrated by non-invasive
detection and multi-way analysis. Skinresearch and Technology, 8, 194-201.
Background: It is known that the properties of human skin vary locally. The purpose
of this study was to investigate the properties of human volar forearms even further
using advanced non-invasive techniques and numerical methods. Methods: The skin
properties of human volar forearms were investigated using measurements of trans
epidermal water loss and multifrequency electrical impedance. Eight sites on the
forearms of 27 healthy volunteers were measured. The sites were oriented as squares,
four sites on both left and right forearm, approximately 40-50 mm apart.Results:
Analysis of variance showed significant differences for epidermal water loss (P < 0.01)
and the skin impedance (P < 0.001) between the inner and outer sides of the arms.
Additionally, parallel factor analysis of the full skin impedance spectra also showed
that there are systematic differences between right and left arm (P < 0.01).Conclusions:
It is crucial to design skin studies carefully in order to minimise the effects of
the local skin properties of human skin.
Acar, E., Çamtepe, S. A., Krishnamoorthy, M. S., & Yener, B. (2005).
Modeling and multiway analysis of chatroom tensors. In ISI 2005: IEEE International Conference on
Intelligence and Security Informatics, Lecture Notes in Computer Science 3495, pp 256-268. Springer Verlag.
This work identifies the limitations of n-way data analysis
techniques in multidimensional stream data, such as Internet chatroom
communications data, and establishes a link between data collection and
performance of these techniques. Its contributions are twofold. First, it
extends data analysis to multiple dimensions by constructing n-way data
arrays known as high order tensors. Chatroom tensors are generated by a
simulator which collects and models actual communication data. The accuracy
of the model is determined by the Kolmogorov-Smirnov goodnessof-
fit test which compares the simulation data with the observed (real)
data. Second, a detailed computational comparison is performed to test
several data analysis techniques including svd , and multiway techniques
including Tucker1, Tucker3 , and Parafac .
Acar, E., Bingol, C. A., Bingol, H., & Yener, B. (2006).
Computational analysis of epileptic focus localization. In Proc. of The Fourth IASTED International
Conference on Biomedical Engineering.
Epilepsy surgergy outcome strongly depends on the localization
of epileptic focus. The analysis of ictal EEG (scalp
or intracranial) is a gold standard for definition of localization
of epileptic focus. In order to automate visual analysis
of large amounts of EEG data, we examine the correlations
among electrodes captured by linear, nonlinear and multilinear
data analysis techniques. We study the performance
of these statistical tools to understand the complex structure
of epilepsy seizure and localize seizure origin. Our analysis
results on four patients with temporal lobe epilepsy reveal
that multiway (Tucker3 ) and nonlinear multiway (Kernelized
Tucker3) analysis techniques are capable of capturing
epileptic focus precisely when validated with clinical
findings whereas linear and nonlinear analysis techniques
(SVD, Kernel PCA) fail to localize seizure origin.
Achim, A., & Bouchard, S. (1997).
Toward a dynamic topographic components model.
Electroencephalography and Clinical Neurophysiology, 103, 381-385.**
Möcks' topographic component model (tcm) (Möcks, J. Topographic components
model for event-related potentials and some biophysical considerations. IEEE Trans.
Biomed. Eng., 1988a, 35: 482-484; Möcks, J. Decomposing event-related potentials:
A new topographic components model. Biol. Psychol., 1988b, 26: 199-215) decomposes
event-related potentials into components uniquely determined by their respective
amplitude profiles across replicates, assuming a constant topography and wave shape
for each component. To accommodate possible changes in the component expression
across conditions, a dynamic version of tcm is investigated which further admits
component modulation in time scale. Twenty test problems were synthesized, incorporating
two arbitrary topographies each activated with its own arbitrary wave shape modified,
across two conditions, in amplitude, onset and duration. Seventeen problems were
perfectly solved, with substantial success on the remaining three, confirming that
component jitter or stretching can even help component identification.
Achim, A., & Marcantoni, W. (1997).
Principal component analysis of event-related potentials: Misallocation of variance
revisited. Psychophysiology, 34, 597-606.**
Misallocating variance, in event-related potential analysis, refers to attributing
an experimental effect to components not actually affected. a vector interpretation
of the relationship between mathematically derived and hue underlying components
shows that misallocation depends exclusively on incorrect identification of the
affected component. Simulations, using seven imperfect rotations, confirmed all
predictions from the vector interpretation concerning the presence, direction, and
size of misallocated variance. Contrary to principal component analysis (pca),
Möcks's topographic component model (tcm) is not subject to rotation problems.
These two methods were compared over 100 simulations in which the components had constant
waveforms and topographies across participants. The group effect was always detected,
but only pca and not tcm showed significance on other components, except when their
random weights happened to differ between groups.
Adamopoulos, J. (1982).
The perception of interpersonal behavior: Dimensionality and importance of the
social environment. Environment and Behavior, 14, 29-44.
This study is aimed towards an analysis of individual cognitive structures with
respect to behavior in the social environment. All analyses were performed on the
deviation scores from the grand mean of all the observations. Each of the ten subjects
produced the following data: 12 behaviors by 12 roles by 8 situations. Tucker's (1966)
three-mode factor analysis was employed, both for females and males and for each of
the ten subjects. After selection of the factors the eigenvector matrices were
rotated to varimax simple structures.
Adamopoulos, J. (1984).
The differentiation of social behavior: Toward an explanation of universal interpersonal
structures. Journal of Cross-Cultural Psychology, 15, 487-508.
The 35 (behaviors) by 6 (resources) by 16 (informants) matrix was analyzed using
Tucker's (1966) three-mode analytic procedure. All analyses were performed on the
deviation scores around the grand mean of all observations. A principal component
analysis of the cross-products yielded five behavior factors. The corresponding
eigenvector matrix was subsequently rotated to varimax simple structure. Two informant
and four resource factors were also rotated using the varimax criterion. The core
matrix was transformed according to these rotations.
Kenmerken van werk. Doctoral thesis. Leiden, The Netherlands: author.
As part of a larger study T3 (as implemented by Kroonenberg & De Leeuw, 1980)
was performed on 25 jobs in a steel factory, 24 tasks and 10 judges to check whether
the judges agreed on the relationships between jobs and tasks. No numerical details
Allosio, N., Boivin, P., Bertrand, D., & Courcoux, P. (1997).
Characterisaion of barley transformation into malt by three-way factor analysis
of near infrared spectra. Journal of Near Infrared Spectroscopy, 5,
Data collected for the spectral study of time series can be presented as a three-way
array in which the three modes are the batches, time and wavelengths. The parallel
factor analysis (PARAFAC) model is relevant for the analysis of three-way data
tables, while the malting process consists of the time transformation of barley into
malt. Furthermore, samples were collected on each day of an industrial malting process
and their near infrared spectra were recorded in diffuse reflectance mode from 1100
to 2500 nm. The time and the wavelength modes associated with the first component
showed that the spectra intensities allowed the classification of samples according
to time. A study of the other loading vectors weighted by their coefficients on the
time-mode explain some phenomena taking place during malting. PARAFAC allowed us
to separate batches acoording to the malting process to which they were submitted.
Batches were also differentiated according to the chemical modification rate that
occurred as expressed by the biochemical analyses results of the final malts. This
work shows that PARAFAC can be a useful tool in the study of time series.
Alsberg, B.K., & Kvalheim, O.M. (1993).
Compression of nth-order data arrays by B-splines. Part 1: Theory. Journal
of Chemometrics, 7, 61-73.
For efficient handling of very large data arrays, pretreatment by compression is
mandatory. In the present paper B-spline methods are described as good candidates
for such data array compression. The mathematical relation between the maximum entropy
method for compression of data tables and the B-spline of zeroth degree is described
together with the generalization of B-spline compression to nth-order data
array tables in matrix and tensor algebra.
Alsberg, B.K.,& Kvalheim,
Compression of three-mode data arrays by B-splines prior to three-mode principal
component analysis. Chemometrics and Intelligent Laboratory Systems, 23, 29-38.
Three-mode PCA is very computer demanding. It requires a large amount of storage
space and many floating point operations (FLOPS). By using three-mode B-spline
compression of three-mode data arrays, the original data array can be replaced by
a smaller coefficient array. Three-mode principal component analysis (PCA) is then
performed on the much smaller coefficient array instead of on the original array.
For the compression approach to be efficient the three-mode data array is assumed
to be well approximated by smooth functions. The smoothness affects the dimensions
of the coefficient array. It is always possible to approximate the data to any
precision but the reward in reduced computation time and storage is lost when the
dimensions of the coefficient array approach dimensions of the original array.
Alsberg, B.K., & Kvalheim, O.M.(1994b).
Speed improvement of multivariate algorithms by the method of postponed basis matrix
multiplication. Part I: Principal component analysis. Chemometrics and Intelligent
Laboratory Systems, 24, 31-42.
Compression is one way of making analysis of large data matrices faster. Compression
is here defined as the case when a alarge matrix X is replaced by a smaller coefficient C.
The coefficients are obtained by least squares fitting to some compression basis. When
performing, e.g., principal component analysis (PCA) of C, the results are comparable
but not equal to the results from analyzing X. In this paper a solution to this problem
is suggested by rewriting the PCA algorithm in terms of C and the compression basis
matrices. This has been accomplished by applying a method where speed improvent is
achieved by postponing basis matrix calculations in key steps of the PCA algorithm.
The method suggested can also be applied to other (but not all kinds of) multivariate
Alsberg, B.K.,& Kvalheim,
Speed improvement of multivariate algorithms by the method of postponed basis matrix
multiplication. Part II: Three-mode principal component analysis. Chemometrics
and Intelligent Laboratory Systems, 24, 43-54.
Compression is one way of making analysis of large data arrays faster. Compression
is here defined as the case when a large array X is replaced by a smaller coefficient
array C. The coefficients are obtained by least squares fitting to some compression
basis. This paper deals with three-mode arrays. When performing, e.g., three-mode
principal component analysis of C the results are comparable but not equal to the
results from analyzing X. In this paper a solution is suggested to this problem by
rewriting the three-mode PCA algorithm which utilizes C and the compression basis
matrices. This has been accomplished by applying a method where speed improvement
is achieved by postponing basis matrix calculations in key steps of the three-mode
Alsberg, B.K. (1997).
A diagram notation for N-mode array equations. Journal of Chemometrics,
An intuitive and user-friendly notation based on a diagrammatical way to visualise
and manipulate N-mode array equations is proposed. The diagrams have the
appearance of graphs and make some manipulations of equations involving higher-order
Altink, W.M.M., & Born, M.Ph. (1987).
Achievement strategies in work organizations: Concept analysis and developement of
a situation-response inventory. Social and Behavioral Science Documents, 17, ##-##.
An SR-questionaire was applied in the context of organisational problems. The methods
of analysis employed here are 1) Three-way analysis of variance and 2)three-mode
principal component analysis (Kroonenberg, 1983). No numerical details given.
Altink, W.M.M., Born, M.Ph., & Algera, J.A. (1987).
De SR-vragenlijstmethode bij organisatievraagstellingen: Nieuwe mogelijkheden voor
diagnostiek van werkgedrag? [The method of SR-inventory in workorganizations: New
possibilities for diagnostics of work-attitude?] Gedrag en Organisatie,
The method of the SR-(situation-response-) inventory is not yet used within
workorganizations. Up until now, its use is restricted to studies within the domain
of personality psychology. A unique characteristic of the SR-inventory is that
respondents describe their response-styles in relation to various situations.
Consequently, it is possible to investigate the relationships between situation-,
response-, and person-factors. In this article the method of the SR-inventory is
discussed with respect to its use in personnel selection and organizational change.
Results obtained with a recently developed SR-inventory for the measurement of
managerial workstyles serve as an illustration. The conclusion is that the method
of the SR-inventory has several advantages over other research-techniques currently
being used to analyze organizational problems. When actually using the SR-inventory
within workorganizations, some aspects need special attention.
Amato, V. (1989).
Autoregressive process for three-way data matrices. In R. Coppi and S. Bolasco (Eds.),
Multiway data analysis (pp. 383-389). Amsterdam: Elsevier.
A structural autoregressive process of first order (Amato, 1967, 1977) is here
extended to the analysis of three-way data matrices describing complex evolutive
phenomena. The proposed approach consists of three steps. The first is to estimate
the basic matrix that generates the process. The second performs the spectral
decomposition associated to this matrix in order to evidence the so called latent
environments. The third step is devoted to asymptotic behaviour study of the series.
A numerical example on real data will conclude the paper.
Amato, V. (1992).
Special vec for a diagonal form of a group of matrices.
Statistica Applicata, 4, 653-657.
This article generalizes the usual method of principal components analysis
of a real square matrix to principal components analysis of a group of
matrices. We shall use the theory of characteristic eigenvalues and
but with reference to several real square matrices
of order n arranged to form a labda-matrix in the Frazer, Duncan &
Collar sense. If the determinant of the polynomial of the highest degree m
with not null matrix coefficients As (s=1,...,m) independent of
labda is zero, then As is said to be a G group. If in particular
m=1, we obtain the usual analysis for the only square matrix A1,
that represents in this special case an improper G group. Partitioned
order nm, called W, that includes As, is treated here.
Associated with W there exists a Vm-1 special vec of nm
components, denominated eigenvec of degree m-1, such that W
Vm-1=Vm. Besides this paper considers
for G, the Flury (1984) hypothesis which positive definite symmetric
matrices As are simultaneously diagonalizable:
B'AsB=Ds (diagonal), with B orthogonal
of order n. A rule to rearrange the nm eigenvalues of W into m diagonal
matrices deltas, is here illustrated. A diagonal form for
G, in the Flury assumption, concludes the article.
Amrheim, M., Srinivasan, B., Bonvin, D., & Schumacher, M. M. (1996).
Inferring concentrations on-line from near-infrared spectra: Nonlinear calibration via
Computers & Chemical Engineering, 20, S975-S980.
Near-Infrared (NIR) spectroscopic methods for
on-line concentration estimation are gaining
popularity in chemical production. The problem
with NIR data, however, is its nonlinear nature
rendering standard factor-analytical (FA)
techniques inapplicable to estimate the
concentrations. To infer concentrations on-line
from NIR spectra, the following steps are
proposed: (i) measure both the NIR and
Mid-Infrared (MIR) spectra of unknown reaction
mixtures in the laboratory (MIR spectroscopy has
the desirable property of linearity with respect
to concentrations), (ii) apply FA techniques to
MIR data and infer concentrations, (iii) calibrate
NIR against the estimated concentrations by using
standard nonlinear regression methods and (iv) use
the calibration model in subsequent production
runs. Experimental results are presented to
illustrate the efficiency of this method.
Andersen, A. H. & Rayens, W. S. (2004).
Structure-seeking multilinear methods for the analysis of fMRI data.
NeuroImage, 22, 728-739.
In comprehensive fMRI studies of brain function, the data structures
often contain higher-order ways such as trial, task condition, subject,
and group in addition to the intrinsic dimensions of time and space.
While multivariate bilinear methods such as principal component
analysis (PCA) have been used successfully for extracting information
about spatial and temporal features in data from a single fMRI run, the
need to unfold higher-order data sets into bilinear arrays has led to
decompositions that are nonunique and to the loss of multiway linkages
and interactions present in the data. These additional dimensions or
ways can be retained in multilinear models to produce structures that
are unique and which admit interpretations that are neurophysiolog-ically
meaningful. Multiway analysis of fMRI data from multiple runs
of a bilateral finger-tapping paradigm was performed using the
parallel factor (PARAFAC) model. A trilinear model was fitted to a
data cube of dimensions voxels by time by run. Similarly, a quadri-linear
model was fitted to a higher-way structure of dimensions voxels
by time by trial by run. The spatial and temporal response components
were extracted and validated by comparison to results from traditional
SVD/PCA analyses based on scenarios of unfolding into lower-order
Andersen, C. M. (2003).
New aspects of chemometrics applied to spectroscopy -
examples from research in fish production.
Unpublished PhD thesis, The Royal Beterinary and Agricultural University, Stockholm.
The overall purpuse of the present Ph.D. thesis is to show
how new as well as known chemometric methods can be applied to
spectroscopic data. The spectroscopic measurements are obtained as
multivariate or multi-way data, which due to complexity of the measured
samples can be rather complex. Therefore, chemometric methods are
necessary in order to resolve and fully understand the underlying
structure reflected in the measurements. Fluorescence spectroscopy and
nuclear magnetic resonance (NMR) measured on fish muscles are used as
example to illustrate the potential use of the methods on real products.
Focus has been on three main subject, which are exemplified in the six
included papers. The three subjects are:
1) Exploratory analysis of
2) Validation of models
3) Quantifying and
handling errors in instrumental measurements
spectroscopic measurements makes it possible to measute the product
non-destructively. However, such measurements can be rather complex due to
the composition and heterogeneity of the food product. This is illustrated
by exploratory analysis of fluorescence emissions of cod and salmon,
fluorescence landscapes of fish muscle extracts and low-field NMR measured
on gadoid fish muscle. Using principal component analysis (PCA) and
parallel factor analysis (PARAFAC) to interpret the data, it is suggested
that fluorescence spectra of fish muscle are primarily due to the
connective tissue, NADH, ocidattion product, lipids, pigments and amino
acids. Slicing modelling of low-field NMR decays shows the distribution of
water within fish fillets and the division of water into different pools.
The content of water in these pools is correated to the water holding
capacity (WHC) of the fish muscle, which is an important quality
parameter. Furthermore, the exploratory analysis shows how three-way data
can be decomposed and interpreted. Problems involved in PARAFAC modelling
of fluorescence excitation-emission data such as missing values and
scatter are discussed and examples of how these can be handled are
illustrated. Sampling is an important aspect to consider, since fish
muscle as other biological material is heterogeneous. An example is given
of how sampling errors in instrumental measurements can be quantified and
handled when these measurements are used for estimating a reference value
by univariate or multivariate regression. Furthermore, it is shown how
traditional univariate statistics can be used in multivariate situations
by calculating the net analyte signal. Validation is another important
factor that should be considered in all data analysis. Validation is
necessary to ensure valid and reliable modelling and is included as a part
of the chemometric modelling in all the papers. In the thesis it is
described how the chosen validation method depends on the purpose of the
data analysis and the chemometric model. Overall, the project has
illustrated that spectroscopic methods such as fluorescence and NMR in
combination with chemometrics can provide information about underlying
chemical and physical parameters in fish. The results are promising and
give indications of considerable potential for the used method, which
encourages further research and development.
Andersen, C. M. & Bro, R. (2003)
Practical aspects of PARAFAC modeling of fluorescence excitation-emission data
Journal of Chemometrics, 17, 200-215.
This paper presents a dedicated investigation and practical description of
how to apply PARAFAC modeling to complicated fluorescence excitation-emission
measurements. The steps involved in finding the optimal PARAFAC model are
described in detail based on the characteristics of fluorescence data. These
steps include choosing the right number of components, handling problems with
missing values and scatter, detecting variables influenced by noise and
identifying outliers. Various validation methods are applied in order to ensure
that the optimal model has been found and several common data-specific problems
and their solutions are explained. Finally, interpretations of the specific
models are given. The paper can be used as a tutorial for investigating
fluorescence landscapes with multi-way analysis.
Anderson, C.J. (1996).
The analysis of three-way contingency tables by three-mode
association models. Psychometrika, 61, 465-483.
The RC(M) association model (Goodman, 1979, 1985, 1991) is useful for analyzing the
relationship between the variables of a 2-way cross-classification. The models presented here
are generalizations of the RC(M) association model for 3-way tables. The family of
models proposed here, "3-mode association" models, use Tucker's 3-mode components model
(Tucker, 1964, 1966; Kroonenberg, 1983) to represent either the three factor interaction or the
combined effects of two and three factor interactions. An example from a study in developmental
psychology (Kramer & Gottman, 1992) is provided to illustrate the usefulness of the
Andersson, C.A. (1999).
Multiway data in chemometrics. Exploratory data
analysis in chemistry with soft multi-linear modelling. Bulletin of the
International Statistical Institute, 58, 215-218.
The most popular models for the analysis of multi-way data structures, i.e., the
Tucker and CANDECOMP-PARAFAC (CP) models, were developed in the domain of
numerical psychology. However, data collected by many modern analytical chemical
instruments are multi-way data structures per definition, and many applications
have shown that data analysis gains in robustness and information outcome by
deriving meaningful parameters from appropriately chosen multi-linear models of
such measurements. The current presentation covers the two basic models of
widest general interest and two applications from the sugar industry. Both
applications are based on fluorescence measurements of aqeous samples.
Andersson, C.A. (2000).
Exploratory multivariate data analysis with applications in food
technology. Kopenhagen, Denmark: DSR Grafik.
Section I: Exploratory multivariate data analysis
Section II: Algorithms, models and applications: 1, direct orthogonalization; 2,
A new criterion for simple structure transformations of core arrays in N-way PCA
with application to fluorometric data; 3, A general algorithm for obtaining
simple structure of core arrays in N-way PCA with application to fluorometric
data; 4, Improving the speed of multiway algorithms. Part I: Tucker3; 5,
Improving the speed of multiway algorithms. Part II: Compression; 6, Further
improvements of the speed of the three-way Tucker3 algorithm; 7, Chemometrics in
food science; 8, Multi-way chemometrics for mathematical separation of
fluorescent colorants and colour precursors from spectrofluorimetry of beet
sugar and beet sugar thick juice as validated by HPLC analysis; 9, Analysis of
N-dimensional data arrays from fluorescence spectroscopy of an intermediate
sugar product; 10, PARAFAC2- Part II. Modeling chromatographic data with
retention time shifts.
Andersson, C.A., & Bro, R. (1998).
Improving the speed of multi-way algorithms: Part I. Tucker3. Chemometrics
and Intelligent Laboratory Systems, 42, 93-103.
In an attempt to improve the speed of multi-way algorithms, this paper
investigates several different implementations of the Tucker3 algorithm.
The interest is specifically aimed at developing a fast algorithm in the MATLAB
(TM) environment that is suitable for large data arrays. Nine
different implementations are developed and tested on real and simulated data.
In a subsequent paper, it will be demonstrated that a fast
algorithm for the Tucker3 model provides a perfect basis for improving the speed
of other multi-way algorithms. From the Internet address
http:\\www.models.kvl.dk\source\, the developed algorithms can be
Andersson, C. A., & Bro, R. (2000).
The N-way Toolbox for MATLAB.
Chemometrics and Intelligent Laboratory Systems, 52, 1-4.
This communication describes a free toolbox for MATLABw for analysis of multiway data.
The toolbox is called ‘‘The N-way Toolbox for MATLAB’’ and is available on the internet at
http:rrwww.models.kvl.dkrsourcer. This communication is by no means an attempt to summarize or review
the extensive work done in multiway data analysis but is intended solely for informing the reader of
the existence, functionality, and applicability of the N-way Toolbox for MATLAB.
Andersson, C.A., & Henrion, R. (1999).
A general algorithm for obtaining simple structure of core arrays in N-
way PCA with application to fluorometric data. Computational Statistics &
Data Analysis, 31, 255-278.
Simplifying the structure of core arrays from N-way PCA or Tucker3 models is
desirable to allow for easy interpretation of the factor estimates. In the
present paper, first a general algorithm for maximizing a differentiable goal
function depending on a set of orthogonal matrices is formulated and then
specified to the problem of estimating orthonormal transformation matrices for
rotating core arrays to simpler structure. The generality of the chosen approach
allows to cope with all possible transformation criteria by just changing one
command in the implementation. In particular, the classical body-and slice-wise
diagonalization of core arrays as well as the recently proposed maximization of
the variance of squared entries are covered. The stability of the algorithm is
addressed by a simulation study using 120 three-way core arrays of dimension
(4,4,4). Each core array instantiates a class of 50 equivalent cores by random
orthonormal transformations. Theoretically, each core within a given class has
the same optimum with respect to the chosen criterion, and the ability of the
algorithm to provide that result has been investigated. The algorithm proves to
work with a high degree of stability and consistency in optimizing the three
discussed goal functions. In addition, theoretical convergence results of the
algorithm are provided. In particular, monotonic convergence of functional
values and convergence of iterates towards a stationary solution an proven. To
illustrate the effect of maximizing the variance-of-squares and the
functionality of the algorithm, the proposed method is applied to a three-way
data array from fluorometric analysis of fractions obtained from low-pressure
chromatographic separation of a preliminary sugar product, thick juice. A
significant gain in simplicity is achieved, and in particular optimizing
variance-of-squares provides a simple core structure for the data under
investigation. The proposed algorithms for maximizing variance-of-squares, body
diagonality and slice-wise diagonality have been implemented in MATLAB and are
available by contact to the authors.
Andersson, C.A., Munck, L., Henrion, R., & Henrion, G. (1997).
Analysis of N-dimensional data arrays from
fluorescence spectroscopy of an intermediary sugar
Fresenius Journal of Analytical Chemistry, 2, 2.
Unwanted formation of colour takes place during
the production of crystalline sugar. The degree of
colouration depends partly on the necessary
processing conditions, e.g. Heating and pH, and
partly on the initial composition and condition of
the sugar beets used as raw material. Reducing
sugars are formed during the process. These are
reactive compounds forming a variety of coloured
complexes and strong precursors to further
formation of colour and many of these compounds
contain fluorophores. In the present work it is
discussed if spectrofluorometric screening of
intermediary sugar products prior to the final
heating stages combined with a multiway
chemometric approach can provide information that
significantly reflects the condition of the
process and the beets. The model used is the n-way
pca (principal component analysis) which is an
exploratory model, not necessitating explicit
modelling of single parameters nor any assumptions
towards parameter interaction. By use of a 4-way
pca of order (3,2,3,3) satisfactory classification
of 47 thick juice samples belonging to 5 factories
has been obtained from a spectrofluorometric
screening method. Also, a temporal trend has been
found to evolve during the time of production. The
investigation substantiates the use of modern
models from data analysis for extracting
significant information from large and complex
Andersson, G.G., Dable, B.K. & Booksh, K.S. (1999).
Weighted parallel factor analysis for calibration
of HPLC-UV/Vis spectrometers in the presence of
Beer's law deviations.
Chemometrics and Intelligent Laboratory Systems, 49, 195-213.
An extension of the parallel factor analysis
(PARAFAC) methodology is presented to allow
accurate and reliable quantitative and
qualitative analysis of nonlinear data collected
from hyphenated instrumentation. The weighted
PARAFAC method is applied to high-performance
(HPLC-UV/Vis) diode array spectrometry analysis.
It is demonstrated that this method improves the
quantitative errors when spectroscopic
nonlinearities from solvent-solute interactions
or detector saturation are introduced. As much as
50% improvements in the root mean squared errors
of estimation are realized for test samples. This
weighted PARAFAC algorithm implicitly treats
nonlinear data as missing values. A method
requiring no a priori information is presented,
that facilitates determination of the nonlinear
regions and optimal application of the weighted
Andersson, M., Josefson, M., Langkilde, F., W., & Wahlund, K. G. (1999).
Monitoring of a film coating process for tablets using near infrared reflectance
Journal of Pharmaceutical and Biomedical Analysis, 20, 27-37.
A process analytical chemical method using near
infrared diffuse reflectance spectrometry was developed for the determination of the
amount of tablet coating on single tablets. This method is based on calibration of the
spectra versus the added mass of coating solution. The tablet core was composed of two
halves of different chemical composition and spectra were recorded from both sides of
the tablets. The calibration was carried out using the chemometric methods principal
component analysis (PCA), partial least squares (PLS), and multiplicative signal
correction (MSC). The PLS-model utilised spectra obtained from both sides, pretreated
with MSG, and ordered into one object. This method can be used in process analytical
chemistry at-line. Additional characterisation of the measurements was obtained by
calibrating the spectra versus coating thicknesses obtained from optical microscopy.
Using PCA, it was possible to roughly estimate the maximum depth in the coating
material that returns chemical information, the 'information depth', which was 0.1-0.2 mm.
Andersson, M., & Knuuttila, K. G. (2002).
The multivariate use of vibrational spectroscopy for chemical characterisation of
Vibrational Spectroscopy, 29, 133-138.
Chromatography media used in purification of biomolecules are extensively
tested with various methods to ensure high quality products. The spherical and porous polymer
particles are derivatised to give different chromatographic properties. The variations in,
for instance, ligand content and chromatographic function are generally tested with a number
of conventional analytical methods (mainly wet-chemical and functional methods) and these are
time consuming. However, by using vibrational spectroscopy, chemical information can be
efficiently obtained from the spectral data. In combination with traditional analytical data,
multivariate models can be constructed. These models can then be used as a tool for
determination (prediction) of, for example, ligand content. In the present work, prediction
of ionic capacity and content of allyl groups are exemplified by Raman spectroscopy of two
different types of agarose-based media (Sepharose(TM) Fast Flow Sepharose(TM)).
The conclusion is that vibrational spectroscopy is a simple, fast and highly informative tool
for both qualitative and quantitative characterisation of adsorbents used in chromatography
Andrews, D. T., & Wentzell, P. D. (1997).
Applications of maximum likelihood principal component analysis: incomplete data
sets and calibration transfer.
Analytica Chimica Acta, 350, 341-352.
The application of a new method to the multivariate analysis of
incomplete data sets is described. The new method, called maximum likelihood principal
component analysis (MLPCA), is analogous to conventional principal component analysis
(PCA), but incorporates measurement error variance information in the decomposition of
multivariate data. Missing measurements can be handled in a reliable and simple manner
by assigning large measurement uncertainties to them. The problem of missing data is
pervasive in chemistry, and MLPCA is applied to three sets of experimental data to
illustrate its utility. For exploratory data analysis, a data set from the analysis
of archeological artifacts is used to show that the principal components extracted by
MLPCA retain much of the original information even when a significant number of
measurements are missing. Maximum likelihood projections of censored data can often
preserve original clusters among the samples and can, through the propagation of error,
indicate which samples are likely to be projected erroneously. To demonstrate its
utility in modeling applications, MLPCA is also applied in the development of a model
for chromatographic retention based on a data set which is only 80% complete. MLPCA can
predict missing values and assign error estimates to these points. Finally, the problem
of calibration transfer between instruments can be regarded as a missing data problem in
which entire spectra are missing on the 'slave' instrument. Using NIR spectra obtained
from two instruments, it is shown that spectra on the slave instrument can be predicted
from a small subset of calibration transfer samples even if a different wavelength range
is employed. Concentration prediction errors obtained by this approach were comparable
to cross-validation errors obtained for the slave instrument when all spectra were
Antunes, A. M., Ferreira, M. M. C., Melgo, M. S., & Volpe, P. L. O. (1999).
Potassium transport through liquid membranes using spectral and chemometrics methods.
Journal of Molecular Structure, 481, 563-567.
Chemometric techniques were applied to follow the transport of potassium
through a chloroform membrane, using ternary mixtures of the isomeric anions,
2,4-dinitrophenolate (2,4-dnp) and 2,5-dinitrophenolate (2,5-dnp) with also 2-nitrophenolate
(ortho-). Some spectral differences can be observed in the uv-vis region, although 2,4- and
2,5-dnp have very similar spectra. Using the trilinear decomposition method (tld), it is
possible to estimate the spectra of the pure components and simultaneously their kinetic
profiles. The estimated spectral profiles are in excellent agreement with experimental results
and the kinetic profiles agree with those obtained by ordinary least squares (ols). The
transport rates are calculated and compared. The ortho- isomer has a very small rate of
Appellof, C.J., & Davidson, E.R. (1981).
Strategies for analyzing data from video fluorometric monitoring of
liquid chromatographic effluents. Analytical Chemistry, 53,
In this paper, the Tucker method is used for qualitative analysis of a
multicomponent fluorescent mixture.
Appellof, C.J., & Davidson, E.R. (1983).
Three-dimensional rank annihilation for multi-component
determinations. Analytica Chimica Acta, 146, 9-14.
The method of rank annihilation for multi-component determinations is extended to a
three-dimensional data array. The possibility of improved sensitivity over the two-dimensional
method is shown. An illustration using data of the type expected from a liquid chromatograph
with a video-fluorimeter as detector is presented.
Arabie, P., & Boorman, S.A. (1973).
Multidimensional scaling of measures of distance between partitions.
Journal of Mathematical Psychology, 10, 148-203.
The techniques of multidimensional scaling were used to study the numerical behavior of twelve
measures of distance between partition lattices of four different sizes. The results offer
additional support for a system of classifying partition metrics, as proposed by Boorman (1970)
and Boorman and Arabie (1972). While the scaling solutions illuminated differences between the
measures, at the same time the particular data with wich the measures were concerned offered a
basis both for counterexamples to some common assumptions about multidimensional scaling and
for some conjectures as to the nature of scaling solutions. The implications of the latter
findings for selected examples from the literature are considered. In addition, the methods of
partition data analysis discussed here are applied to an example using sociobiological data.
Finally, an argument is amde against undue emphasis upon interpreting dimensions in nonmetric
Arabie, P., & Carroll, J.D. (1980b).
MAPCLUS: A mathematical programming approach to fitting the ADCLUS
model. Psychometrika, 45, 211-235.
We present a new algorithm, MAPCLUS (MAthematical Programming
CLUStering), for fitting the Shepard-Arabie ADCLUS (for ADditive
CLUStering) model. MAPCLUS utilizes an alternating least squares method
combined with a mathematical programming optimization procedure based on a
penalty function approach, to impose discrete (0,1) constraints on parameters
defining cluster membership. This procedure is supplemented by several other
numerical techniques (notably a heuristically based combinatorial optimization
procedure) to provide an efficient general-purpose computer implemented
algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an
application to one of the examples given by Shepard and Arabie using the older
ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve
nearly the same level of goodness-of-fit. Finally, we consider an extension of
the present approach to fitting a three-way generalization of the ADCLUS model,
called INDCLUS (INdividual Differences CLUStering).
Arabie, P., Carroll, J.D., DeSarbo, W.S., & Wind, J. (1981).
Overlapping clustering: A new method for product positioning. Journal of Marketing
Research, 18, 310-317.
Most clustering techniques used in product positioning and market segmention studies
render mutually exclusive equivalence classes of the relevant products or subject
space. Such classificatory techniques are thus restricted to the extent that they
preclude overlap between subsets or equivalence classes. An overlapping clustering
model, ADCLUS, is described which can be used in marketing studies involving
products/subjects that can belong to more than one group or cluster simultaneously.
The authors provide theoretical justification for and an application of the approach,
using the MAPCLUS algorithm for fitting the ADCLUS model.
Arabie, P., & Maschmeyer, C. (1988).
Some current models for the perception and judgment of risk.
Organizational Behavior and Human Decision Processes,
We survey some of the models more recently used to portray panelists'
perceptions of risk, viewed as a complex psychological response. These models
are compared (a) as continuous versus discrete, (b) with regard to type of data
and tasks required of panelists, and (c) by facility for portraying different
patterns of judgments among panelists. Substantive results from applying
different models to the same risks and/or data are presented. Finally, we
consider possible future directions for research in the perception of risk,
oriented toward use of the models presented here.
Arabie, P., Mashmeyer, C.J., & Carroll, J.D. (1987).
Impact scaling: Method and application. Technological Forecasting
and Social Change, 32, 245-272.
Impact scaling is a method of forecasting, using expert panelists'
conditional and unconditional probabilities of selected possible future
well as the events' likely effects on such indicators as business
the recent availability of a discrete psychological model (INDCLUS) for
representing structure of the events as well as differences among the
more elaborate and versatile set of models for implementing impact
demonstrated. An illustration is provided using panelists of varied
status to give judgments about impacts of possible future events on
of stock market performance. Finally, further extensions of impact
enhance the utility and precision as a model of judgment and method of
forecasting is considered.
Arancibia, J. A., Olivieri, A. C., & Escandar, G. M.(2002).
First- and second-order multivariate calibration applied to biological
samples: determination of anti-inflammatories in serum and urine.
Analytical and Bioanalytical Chemistry, 374, 451-459.
Two different spectrofluorimetric methods for the determination of
piroxicam (PX) in serum are presented and discussed. One of them is based on the use of
three-way fluorescence data and multivariate calibration performed with parallel factor
analysis (PARAFAC) and self-weighted alternating trilinear decomposition (SWATLD). This
methodology exploits the so-called second-order advantage of the three-way data, allowing
to obtain the concentration of the studied analyte in the presence of any number of
uncalibrated (serum) components. The method was developed following two different
procedures: internal standard addition and external calibration with standard solutions,
which were compared and discussed. The second approach investigated is based on the
combination of solid-phase extraction (SPE) and room temperature fluorimetry. Both methods
here presented yield satisfactory results. The concentration range in which PX could be
determined in serum was 1-10 mug ml(-1). The limits of quantification for the experimental
solutions using the chemometric approach were 0.09 mug ml(-1) for the standard addition
mode and 0.12 mug ml(-1) using external calibration (both for PARAFAC and SWATLD
algorithms). In the solid-surface fluorimetric method, the calibration graph was linear up
to 0.22 mug ml(-1) and the limit of quantification was 0.02 mug ml(-1).
Arancibia, J. A., & Escandar, G. M.(2003).
Two different strategies for the fluorimetric determination of piroxicam in serum.
Talanta, 60, 1113-1121.
First- and second-order multivariate calibration of fluorescence
data have been compared as regards the determination of anti-inflammatories and
metabolites in the biological fluids serum and urine. The simultaneous resolution
of naproxen-salicylic acid mixtures in serum and naproxen-salicylic acid-salicyluric
acid mixtures in urine was accomplished and employed for a discussion of the
relative advantages of the applied chemometric tools. The analysis of second-order
fluorescence excitation-emission matrices was performed using iteratively reweighted
generalized rank annihilation method (IRGRAM), parallel factor analysis (PARAFAC),
and self-weighted alternating trilinear decomposition (SWATLD). The results were
compared with first-order fluorescence emission data analyzed with partial least-squares
regression (PLS). In all cases, the performance of the methods was improved through
the formation of inclusion complexes of the analytes with beta-cyclodextrin. The
concentration ranges in which the analytes could be determined were as follows:
naproxen, 0-250 ng mL(-1) in serum and 0-200 ng mL(-1) in urine; salicylic acid,
0-500 ng mL(-1) in serum and 0-300 ng mL(-1) in urine, and salicyluric acid, 0-300
ng mL(-1) in urine.
Arcelay, A.R., Ross, R.T., & Ezzeddine, B.M. (1988).
Photosystem I generates a free-energy change of 0.7 electron
volts or less. Biochimica et Biophysica Acta - Bioenergetics, 936,
The intensity of delayed luminescence from wild-type Scenedesmus
obliquus was used to
refine a previous determination (Marchiarullo, M.A. and Ross, R.T. (1981)
Biochim. Biophys. Acta
636, 254-257) of the free-energy change in Photosystem II, which we find
to be 0,99 eV under
physiological conditions. Similar measurements were made of the very weak
luminescence from a
mutant deficient in plastoquinone A, originally with the expectation that
this luminescence would
be from Photosystem I. However, the action spectrum for excitation of
mutant luminescence is that
of Photosystem II. There is no measurable change in the properties of the
mutant luminescence with
changes in excitation wavelength extending to 708 nm, leading us to
conclude that Photosystem I
contributes less that 5% of the emission 10ms after excitation. This limit
on the intensity of
emision from Photosystem I places an upper limit of 0.69 eV on its free-
Artyushkova, K., & Fulghum, J. E.(2001).
Identification of chemical components in XPS spectra and images
using multivariate statistical analysis methods.
Jounral of Electron Spectroscopy and Related Phenomena, 121, 33-55.
A variety of data analysis methods can be used to
enhance the information obtained from a measurement, or to simplify
extraction of significant components from large data sets. Much work
is needed to improve the quantification and interpretation of XPS
spectra and images from complex organics. Multivariate analysis (MVA)
is increasingly used for applications in electron spectroscopy to aid
the analyst in interpreting the vast amount of information yielded by
spectroscopic techniques. In general, the goals of MVA are to determine
the number of components present, identify the chemical components, and
quantify component concentrations in the mixture. Principal component
analysis (PCA) is frequently used to determine the number of mathematical
components which describe the data set. These mathematical components
must then be related to chemically meaningful components. Various
approaches to solve rotational ambiguities of spectral resolution,
including local rank method (EFA), pure variables method (Simplisma)
and multivariate curve resolution (MCR), are tested in the determination
of chemical components from XPS data. Limitations associated with the
resolution of a single matrix are shown to be partially or completely
overcome when several related matrices are treated together. The test
data sets contain XPS images or spectra acquired from blends of
poly(vinyl chloride), PVC, and poly(methyl methacrylate), PMMA. The PVC
degrades rapidly upon exposure to the X-ray beam. Spectra and images
from the blend, acquired as a function of time, provide the multi-dimensional
data sets for algorithm evaluation. In addition to spectral resolution,
multivariate image analysis methods, such as principal component analysis,
are used to extract maps of the pure components from an images-to-spectra
Artyushkova, K., & Fulghum, J. E.(2002).
Multivariate image analysis methods applied to XPS imaging data sets.
Surface and Interface Analysis, 33,185-195.
Recent improvements in imaging photoelectron spectroscopy enhance
lateral and vertical characterization of heterogeneous samples at the cost of
increasing complexity in the XPS data sets acquired. These imaging data sets require
more sophisticated analysis methods than visual inspection if the data are to be
interpreted effectively. Multivariate analysis (MVA) methods are increasingly utilized
in surface spectroscopies to aid the analyst in interpreting the vast amount of
information resulting from these multidimensional data set acquisitions.
In this work, image processing analysis methods are tested on XPS data sets acquired
from polymer blends. Images from the blends, acquired as a function of composition,
time or energy, provide multidimensional data sets for algorithm evaluation. Multivariate
image analysis (MIA) methods such as scatter diagrams, principal component analysis (PCA)
and classification methods are used to extract maps of pure components from degradation and
images-to-spectra data sets. In some cases the MVA results can be compared directly with the
XPS spectra or images, which provide a critical reference point. This work will demonstrate
that additional information can result from the application of MIA methods, even when direct
spectral or image interpretation is possible.
Aruga, R., Gastaldi, D., Negro, G., & Ostacoli, G. (1995).
Pollution of a river basin and its evolution with
time studied by multivariate statistical analysis.
Analytica Chimica Acta, 310, 15-25.
A set of quantitative analytical data for 13
rivers belonging to the basin of the Tanaro
(Piedmont region, north-western Italy) has been
processed by multivariate statistical techniques.
The original matrix, which refers to the year
1990, consisted of 20 chemical and
physico-chemical variables, determined at 44
sampling sites. The following methods have been
used for the treatment of the data: Cluster
analysis (unsupervised pattern recognition),
factor analysis, variable selection by fisher
weights. A comparison has also been made between
the data referring to 1990 and to 1978 in order to
investigate the evolution of the environmental
situation for the Tanaro basin after such a lapse
of time. This comparison has been carried out by
processing the 3-d matrix which collects the data
of the two different periods.
Atkinson, A.C., & Riani, M.(2004).
The forward search and data visualisation.
Computational Statistics, 19, 29-54.
A statistical analysis using the forward search produces many graphs. For
multivariate data an appreciable proportion of these are a variety of plots of the
Mahalanobis distances of the individual observations during the search. Each unit, originally
a point in v-dimensional space, is then represented by a curve in two dimensions connecting
the almost n values of the distance for each unit calculated during the search. Our task is
now to recognise and classify these curves: we may find several clusters of data, or outliers
or some unexpected, non-normal, structure. We look at the plots from five data sets.
Statistical techniques include cluster analysis and transformations to multivariate normality.
Azubel, M., Fernández, F.M., Tudino, M.B. & Troccoli, O.E.
Novel application and comparison of multivariate
calibration for the simultaneous determination of
Cu, Zn and Mn at trace levels using flow
injection diode array spectrophotometry.
Analytica Chimica Acta, 398, 93-102.
Three different calibration approaches: Partial
Least Squares, Unfold Partial Least Squares and
n-way partial least squares are compared in terms
of explained variance, root mean square error of
calibration and root mean square error of
cross-validation. Attention is also focused on
the application of genetic algorithms to spectral
data as a way to obtain an improvement in
calibration accuracy. Influence of initial
starting conditions for the genetic algorithms
(population size, mutation probability, % initial
terms etc.) was investigated by means of
factorial experimental designs.
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en Gezinspedagogiek - Datatheorie |
Centre for Child and
Family Studies |
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The Three-Mode Company |
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First version : 12/02/1997;