Three-Mode Abstracts, Part P
With one can go to the index of
this part of the Abstracts, with
one can go to other
parts (letters) of the Abstracts.
|Pa | Pb |
Pc | Pd |
Pe | Pf |
Pg | Ph |
Pi | Pj |
Pk | Pl |
Pm | Pn |
Po | Pp |
Pq | Pr |
Ps | Pt |
Pu | Pv |
Pw | Px |
Py | Pz |
Paatero, P. (1997a).
Least squares formulation of robust non-negative factor analysis.
Chemometrics and Intelligent Laboratory Systems, 37, 23-35.
Positive matrix factorization (PMF) is a recently
published factor analytic technique where the left
and right factor matrices (corresponding to scores
and loadings) are constrained to non-negative
values, The PMF model is a weighted least squares
fit, weights based on the known standard
deviations of the elements of the data matrix. The
following aspects of PMF are discussed in this
work: (1) Robust factorization (based on the Huber
influence function) is achieved by iterative
reweighting of individual data values. This
appears especially useful if individual data
values may be in error. (2) Desired rotations may
be obtained automatically with the help of
suitably chosen regularization terms. (3) The
algorithms for PMF are discussed. A synthetic
spectroscopic example is shown, demonstrating both
the robust processing and the automatic rotations.
Paatero, P. (1997b).
A weighted non-negative least squares algorithm for three-way "PARAFAC"
factor analysis. Chemometrics and Intelligent Laboratory Systems,
A time-efficient algorithm PMF3 is presented for solving the three-way
PARAFAC (CANDECOMP) factor analytic model. In contrast to the usual
alternating least squares, the PMF3 algorithm computes changes to all
three modes simultaneously. This typically leads to convergence in
40 .. 100 iteration steps. The equations of the weighted multilinear
least squares fit are given. The optional non-negativity is achieved
by imposing a logarithmic penalty function. The algorithm contains a
possibility for dynamical reweighting of the data during the iteration,
allowing a robust analysis of outlier-containing data. The problems
typical of PARAFAC models are discussed (but not solved): multiple local
solutions, degenerate solutions, non-identifiable solutions. The
question of how to verify the solution is discussed at length.
Paatero, P. (1999).
The multilinear engine - a table-driven, least
squares program for solving multilinear problems,
including the n-way parallel factor analysis
Journal of Computational and Graphical Statistics, 8, 854-
A technique for fitting multilinear and
quasi-multilinear mathematical expressions or
models to two-, three-, and many-dimensional data
arrays is described. Principal component analysis
and three-way PARAFAC factor analysis are
examples of bilinear and trilinear least squares
fit. This work presents a technique for
specifying the problem in a structured way so
that one program (the Multilinear Engine) may be
used for solving widely different multilinear
problems. The multilinear equations to be solved
are specified as a large table of integer code
values. The end user creates this table by using
a small preprocessing program. For each different
case, an individual structure table is needed.
The solution is computed by using the conjugate
gradient algorithm. Non-negativity constraints
are implemented by using the well-known technique
of preconditioning in opposite way for slowing
down changes of variables that are about to
become negative. The iteration converges to a
minimum that may be local or global. Local
uniqueness of the solution may be determined by
inspecting the singular values of the Jacobian
matrix. A global solution may be searched for by
starting the iteration from different
pseudorandom starting points. Application
examples are discussed-for example, n-way
PARAFAC, PARAFAC2, linked mode PARAFAC, blind
deconvolution, and nonstandard variants of these.
Paatero, P. (2000).
Construction and analysis of degenerate PARAFAC models. Journal of
Chemometrics, 14, 285-299.
A mathematical framework is presented for constructing degenerate
CANDECOMP/PARAFAC models. It is possible to construct degenerate arrays which
can be approximated by two-factor models to arbitrary precision but which do not
possess an exact two-factor representation. Equivalence of different degenerate
presentations is demonstrated. By using this model, tasks are constructed where
the straight path from the specified starting point to the best-fit solution
will pass through a degenerate area. Swamp behaviour is observed when such tasks
are solved by various algorithms.
Paatero, P. & Andersson, C.A. (1999).
Further improvements of the speed of the Tucker3
Chemometrics and Intelligent Laboratory Systems, 47, 17-20.
An improvement to the Andersson-Bro (A-B)
alternating least squares (ALS) algorithm for the
Tucker3 three-way model is presented. The
published A-B algorithm deals cyclically with the
three modes of the problem. In each ALS substep,
the whole array is projected onto a different
mode. The projections are the dominating
workload. In the improved version, each
whole-array projection is utilized for two ALS
substeps. The same ALS steps are performed as
before but the number of full-sized projections
is cut to half. This almost doubles the speed of
the algorithm without changing its convergence
properties. The possibility to utilize each
full-sized projection for more than two ALS
substeps is discussed.
Paatero, P., & Hopke, P.K. (2002).
Utilizing wind direction and wind speed as independent variables in multilinear
receptor modeling studies. Chemometrics and Intelligent Laboratory Systems,
The problem of identifying sources of airborne pollutants and providing quantitative
estimates of the contributions of each of those sources is important for airborne
particulate matter. Various forms of factor analysis have been applied to this
problem. However, in factor analysis, there is the fundamental problem of
rotational ambiguity that makes the problem ill-posed. Thus, the incorporation
of additional information can be useful in improving the solutions. Especially
for identifying local sources, wind data (direction and speed) could be valuable
additional information in such receptor modeling. However, wind data cannot
be used directly as dependent variables in factor analytic modeling because
the dependence of observed concentrations on wind variables is far from linear.
An expanded multilinear model has been developed in which the wind direction,
speed and other variables are included as independent variables. For each
source, the analysis computes a directional profile that indicates how much
of the concentrations are explained by the factors depending on wind direction,
speed, and other values. This model has been tested using simulated data developed
by the U.S. Environmental Protection Agency as part of a workshop to test
advanced factor analysis methods. For most of the local sources, well-defined
directional profiles were obtained.
Paatero, P., & Hopke, P.K. (2003).
Transforming 3-way factor analytic models so that computational workload
is reduce while noise level in computed factors is not increased.
This work is based on a recent paper by Paatero and Hopke (2003)where it
is shown that omitting high-noise columns from a 2-way matrix can improve
the factor analytic recovery of signal amidst high levels of noise. This
work shows how three-way factor analytic models (PARAFAC or Tucker T3, say)
can be transformed so that signal is concentrated in one part of the array,
while the remaining part contains so little signal that it is advantageous
to omit the remaining part entirely from the analysis. The method is best
applicable to arrays with some degree of continuity along at least one axis.
Thus many kinds of chemometrics arrays can be transformed, while many
environmental arrays cannot.
The transformations have two goals: decrease the computational workload and
possibly also to increase the signal-to-noise ratio (S/N) of the computed
factors. In the simplest situations, the transformations can be implemented
as a pre-processing step in combination with conventional ALS-based routines.
All situations can be handled by using the multilinear engine (Paatero 1999).
Conceptually, the implementation of the method is based on orthogonal Givens
(or Jacobi) rotations in the following way. Givens rotations are performed
between the slices of a three-way array so that the three-way factorization
problem stays invariant. Of course, the appearances of the factors will
change but these changes can be undone afterwards by applying the inverse
transformations. The rotations are arranged so that certain slices of the
array will contain smaller and smaller values, while other slices wil grow.
Overall, the sum of squares of array elements stays constant. Rotations can
be performed in all three dimensions. In this way, part of the array is
emptied of useful information so that practically only noise remains in the
emptied part. In such a situation, S/N may possibly be improved when the
emptied part is omitted from further calculations. The model (PARAFAC, say)
is then only fitted to the remaining part of the array, which is usually
not of rectangular shape. The computational gain will depend on the number
of rejected values. Reduction of work by more than 80% appears possible
in especially favorable cases. The largest computational advantage can be
expected from " fat" arrays, i.e. arrays where all three dimensions
In practice, Givens rotations will probably not be used because they may
require an excessive amount of time. Simpler methods are expected to give
the same advantage.
Paatero, P., Hopke, P.K., Song, X.-H., & Ramadan, Z. (2002).
Understanding and controlling rotations in factor analytic models. Chemometrics
and Intelligent Laboratory Systems, 60, 253-264.
Positive Matrix Factorization (PMF) is a least-squares approach for solving
the factor analysis problem. It has been implemented in several forms. Initially,
a program called PMF2 was used. Subsequently, a new, more flexible modeling tool,
the Multilinear Engine, was developed. These programs can utilize different
approaches to handle the problem of rotational indeterminacy. Although both
utilize non-negativity constraints to reduce rotational freedom, such constraints
are generally insufficient to wholly eliminate the rotational problem. Additional
approaches to control rotations are discussed in this paper: (1) global imposition
of additions among "scores" and subtractions among the corresponding "loadings"
(or vice versa), (2) constraining individual factor elements, either scores
and/or loadings, toward zero values, (3) prescribing values for ratios of
certain key factor elements, or (4) specifying certain columns of the loadings
matrix as known fixed values. It is emphasized that application of these
techniques must be based on some external information about acceptable or
desirable shapes of factors. If no such a priori information exists, then
the full range of possible rotations can be explored, but there is no basis
for choosing one of these rotations as the "best" result. Methods for estimating
the rotational ambiguity in any specific result are discussed.
Paatero, P., & Juntto, S. (2000).
Determination of underlying components of a cyclical time series by means of
two-way and three-way factor analytic techniques. Journal of Chemometrics,
A technique is presented for determining the underlying components in a cyclical
time series which is influenced by one prominent cycle (the diurnal or the
yearly cycle). The separation of the components is based on their different
shapes within this period, assuming that the shape of each component stays
approximately constant with time and that the amplitude of each component is a
slowly changing function. The series is folded into matrix shape so that each
cycle forms one column. The matrix is factorized by principal component analysis
or by positive matrix factorization (non-negatively constrained factor analysis
with individual weighting of data values), resulting in the shape and amplitude
functions for the underlying components Synthetic two-way demonstration examples
are analyzed. As a real-life example, traffic-induced carbon monoxide
concentrations in urban air are analyzed. The CO has a diurnal concentration
cycle which changes shape on weekends. This behavior is explained by two
factors, identified with work-related and other traffic. The CO data in fact
contain another multiplicative cycle, the weekly workdays/weekend pattern.
Arranging the data according to time of day, day of week, and week of the year
creates a three-way array. The method is extended to the analysis of such
arrays. Existing software for the well-known PARAFAC model is used for solving
the three-way model. Two factors are again obtained, Their diurnal and weekly
cycles correspond to the work-related and weekend-related traffic patterns.
Analysis of cyclical multivariate data is discussed: such data are also governed
by the three-way PARAFAC model. The advantage of the PARAFAC model relative to
customary two-way methods is emphasized: there is usually no rotational
ambiguity in PARAFAC results.
Pamment, P. (1987).
Difficulty in assertion: An invariant factors structure
analysis. PhD Thesis. University of Queensland.
This thesis examines assertive communication. The major focus is on the
assessment of difficulty in assertion, and the generalization of assessment over
situations and population samples. The approach described in this research examines difficulty in
assertion within the framework of a general cognitive theory, Scheme Theory (Eckblad, 1981a).
An invariant multi-mode factor analysis technique (McDonald, 1984) is employed to
reflect accurately both positive and negative assertive behaviours and close and distant
Papagni, E. (1992).
High-technology exports of EEC countries - persistence and diversity of specialization
Applied Economics, 24, 925-933.
High technology export of EEC countries is considered. The
analysis relies on a bundle of products selected from the
Eurostat database COMEXT. High technology goods are considered
as an innovative output indicator. A test of the hypotheses of
hysteresis and diversity of trade patterns at country level has
been performed to verify some claims made by the `evolutionary'
theory of innovation and trade. The three-mode principal
component analysis carried out confirms the persistence of
specialization patterns of each EEC country in high technology
export, and highlights their sharp differences.
Pardo, R., Helena, B. A., Cazurro, C., Guerra, C., Deban, L, Guerra, C. M., & Vega, M. A. (2004).
Application of two- and three-way principal component analysis to the interpretation of chemical fractionation results obtained by the use of the BCR procedure
Analytica Chimica Acta, 523, 125-132.
This paper describes as two- and three-way principal component analysis
(PCA) can be advantageously used to interpret the results originating from the B.C.R.
fractionation scheme, an operationally defined speciation procedure used to study the
availability and mobility of trace elements and heavy metals present in environmental
solid samples. The application of this procedure to find the fractionation of 11 trace
elements and heavy metals (Al, As, Cd, Cr, Cu, Mn, Mo, Ni, Pb, V and Zn) in 13 sediments
collected at the Mejillones del Sur bay (Antofagasta, Chile) generated a three-dimensional
data set X (13 x 11 x 4). Although classical two-way PCA applied to the unfolded X-b
matrix (52 x 11) can yield valuable information about the pattern of fractionation of
the chemical elements amongst the B.C.R. fractions, a more profound insight can be found
when applying three-way PCA procedures, such as Parafac, that take into account the true
tri-dimensional structure of the data set. In this study, Parafac has allowed to classify
the sediments according to their associated environmental hazard by means of the A-mode
loadings of two significant factors. By plotting these factors into the physical space
of the Mejillones del Sur bay, the most hazardous areas of the bay have been located.
Pastore, A. (1992).
Some observations on similarity and lack of duality in
interstructure-compromise-intrastructure analysis. Statistica
Applicata, 4, 679-691.
The problem of the chioce of Ok operators
associated with the
occasions, in interstructure-compromise-intrastructure methods is
in the case of K matrices Xk units per variables,
operators Wk and covariance matrices
be used. A comparison of the solutions obtained in the two cases
Ok=Rk is proposed using
indices between interstructures, between units representations and between
representations. An illustrative example is presented concerning the
evolution of the Italian enterprises.
Pedersen, F., Bengtsson, E., & Nordin, B. (1995).
An extended strategy for exploratory multivariate image-anaylis including noise considarations.
Journal of Chemometrics, 9, 389-409.
Multivariate image analysis (MIA) is a powerful tool for many image segmentation and classification problems,
but the interpretation and understanding of the original and resulting multidimensional (multivariate) data are not always
easy. A strategy for MIA has been proposed which describes its usage on multivariate images for segmentation tasks. MIA
starts with principal component analysis (PCA) and then continues with interactive analysis of the output from PCA. In this
paper a number of extensions to MIA are proposed. The extensions are the suggestion to incorporate preprocessing of the
multivariate image in MIA, the suggestion to use synthetic multivariate image models which create a clear-cut situation, and
new visualization tools which improve the interactivity and understanding of the results. Extended MIA is applied on synthetic
multivariate image data simulating a possible application with large noise, positron emission tomography (PET). As a result of
the interactive analysis, suggestions for preprocessing emerge. The developed methodology for handling the noise is then applied
on real PET image data with good results.
Pedersen, F., Bergstrom, M., Bengtsson, E., & Langstrom, B. (1994).
Principal component analysis of dynamic positron emission tomography images.
European Journal of Nuclear Medicine, 21, 1285-1292.
Multivariate image analysis can be used to analyse multivariate medical images. The purpose could be to visualize
or classify structures in the image. One common multivariate image analysis technique which can be used for visualization
purposes is principal component analysis (PCA). The present work concerns visualization of organs and structures with different
kinetics in a dynamic sequence utilizing PCA. When applying PCA on positron emission tomography (PET) images, the result is
initially not satisfactory. It is illustrated that one major explanation for the behaviour of PCA when applied to PET images
is that it is a data-driven technique which cannot separate signals from high noise levels, With a better understanding of
the PCA, gained with a strategy of examining the image data set, the transformations, and the results using visualization tools,
a surprisingly easily understood be derived. The proposed enhance clinically interesting information in a dynamic PET imaging
sequence in the first few principal component images and thus should be able to aid in the identification of structures for
Pedersen, D. K., Munck, L. & Engelsen, S. B. (2002).
Screening for dioxin contamination in fish oil by PARAFAC and N-PLSR analysis of
Journal of Chemometrics, 16, 451-460.
A preliminary investigation of fish oils demonstrates that fluorescence excitation-
emission landscapes evaluated by three-way chemometric methods may be a candidate
for an inexpensive screening method to indicate the level of contamination by dioxins
and PCBs, which are normally analysed by expensive and time-consuming physicochemical
separation techniques such as GC-MS. Fluorescence landscapes of 88 fish oils have
been investigated and showed great variation due to species, season and treatment,
depicting a variation in natural fluorescent components. The fluorescence landscapes
were analysed by PARAFAC. Samples with similar fluorescence fingerprints were selected
from a PARAFAC score plot, and local significant prediction models with PARAFAC/MLR,
N-PLSR and PLSR were established with correlation coefficients in the range from
r = 0.69 (n =10) to r = 0.97 (n = 75) for dioxin and of r = 0.92 (n =12) for PCB.
Application of PARAFAC/MLR and N-PLSR to fluorescence landscapes of fish oils resulted
in local regression models for dioxin determination with prediction errors below
1 ng kg(-1), which is comparable to the reference method. In the PARAFAC model, two
of the modes give the excitation and emission spectra of the pure underlying fluorophores,
while the third mode gives their individual concentrations. Excitation and emission
optima for three or four PARAFAC components in each data set were identified, representing
both positive and negative (quenching) correlation components. It is hypothesized
that the quenching correlation may be effected by the joint contribution of chlorinated
organic compounds in the fish oil, including dioxins and PCBs. Other explanations for
the results are discussed.
Pedersen, H. T., Bro, R., & Engelsen, S. B. (2002).
Towards rapid and unique curve resolution of low-field NMR relaxation data: trilinear
SLICING versus two dimensional curve fitting.
Journal of Magnetic Resonance, 157, 141-155.
In this work an alternative method, named SLICING, for two-dimensional
and noniterative T2 decomposition of low-field pulsed
NMR data (LF-NMR) is proposed and examined. The method
is based on the Direct Exponential Curve Resolution Algorithm
(DECRA) proposed by W. Windig and A. Antalek (1997, Chemom.
Intell. Lab. Syst. 37, 241–254) and takes advantage of the fact that
exponential decay functions, when translated in time, retain their
characteristic relaxation times while only their relative amounts
or concentrations change. By such simple translations (slicing) it
is possible to create a new “pseudo” direction in the relaxation
data and thus facilitate application of trilinear (multiway) data-analytical
methods. For the application on LF-NMR relaxation
data, the method has two basic requirements in practice: (1) two or
more samples must be analyzed simultaneously and (2) all samples
must contain the same qualities (i.e., identical sets of distinct T2 val-ues).
In return, if these requirements are fulfilled, the SLICING (tri-linear
decomposition) method provides very fast and unique curve-resolution
of multiexponential LF-NMR relaxation curves and, as a
spin-off, calibrations to reference data referring to individual proton
components require only scaling of the resulting unique concentra-tions.
In this work the performance of the SLICING method (in-cluding
multiple slicing schemes) is compared to a traditional two-dimensional
curve fitting algorithm named MATRIXFIT through
application to simulated data in a large-scale exhaustive experi-mental
design and the results validated by application to two small
real data sets. Finally a new algorithm, Principal Phase Correction
(PPC) based on principal component analysis, is proposed for phase
rotation of CPMG quadrature data, an important prerequisite to
optimal SLICING analysis.
Pell, R. J. (2000)
Multiple outlier detection for multivariate calibration using robust statistical
Chemometrics and Intelligent Laboratory Systems, 52, 87-104.
Outliers that are incorporated into a multivariate calibration model can significantly reduce the
performance of the model. In the case of multiple outliers, the standard methods for outlier detection can fail to
detect true outliers and even mistakenly identify good samples as outliers. Robust statistical methods are less
sensitive to outliers and can provide a powerful tool for the reliable detection of multiple outliers. This paper
examines the use of robust principal component regression (PCR) and iteratively reweighted partial least squares
(PLS) for multiple outlier detection in an infrared spectroscopic application.
Pereira, E. R., Poppi, R. J., & Arruda, M. A. Z.(2001)
Exploratory analysis of micographic Teflon images.
Mikrochimica Acta, 136, 55-60.
Some analytical techniques are able to produce images, making if possible to obtain a large amount of
information. Working with images, the pixels can be treated like objects in a data matrix. In this way, a multi-way PCA of
micrograph images was applied to the internal surface of a Teflon coil used for microwave sample digestion. This Teflon coil
was used for 120h and had been attacked with different acids. It was cut into 3 pieces (namely: initial, medium and central
portions) in order to obtain 7 micrographs (2 for initial, 2 for medium and 3 for the central part). A micrograph of a piece
from a new Teflon coil was used for comparison. With these 8 micrographs, it was possible to establish a three-dimensional
arrangement. After multi-way PCA was applied to the Teflon surface images, was possible to group a great amount of information
and to characterise different parts of this coil.
Pereira, E. R., Sena, M. M. Arruda, M. A. Z. & Poppi, R. J. (2003)
Exploratory analysis of L'vov platform surfaces for electrothermal atomic absorption
spectrometry by using three-way chemometric tools.
Analytica Chimica Acta, 495, 177-193.
An exploratory analysis of the effects of permanent (Zr) and conventional (Mg and
a mixture of Pd+Mg) chemical modifiers on L'vov platforms for electrothermal atomic
absorption spectrometry (ETAAS) was performed in this work by using two chemometric
tools. Twelve L'vov platforms were used for Al, Cd and Pb determination in biological
slurry samples. For each platform, a different combination of chemical modifier,
number of heating cycles and determined analyte was employed. The morphology of
the platforms was evaluated using scanning electron microscopy (SEM) and the obtained
data was analysed with image principal component analysis (image PCA). The results
allowed differentiating the platform treated with Mg from the other platforms.
In addition, eight residual species (P, S, Ca, Ti, Fe, Zr, Hf and Pd) distributions
were obtained by using micro synchrotron radiation X-ray fluorescence (muSRXRF)
in the same platforms for 100 points across the horizontal axis. These data were
modelled with orthogonal constrained PARAllel FACtor analysis (PARAFAC) and a global
characterisation of the platforms was achieved. Four platforms presented a particularly
behaviour, being so different among themselves, that it was necessary one specific
factor (1-4) to model each one. Factor 5 demonstrated that two platforms presented
a similar behaviour characterised by high Ca and, to a lesser extent, Ti content.
Finally, factor 6 modelled the correlated behaviour of five platforms characterised
by Zr content. It was also observed that the platform morphology has a good connection
with the residual species found in its surface. This kind of study can open a vast
field to exploratory analysis of L'vov platforms in ETAAS. (C) 2003 Elsevier B.V.
All rights reserved.
Pernin, M.-O. (1986).
Contribution à la méthodologie d'analyse de données
longitudinales: Exemple de la croissance chez l'être humain (Auxologie).
Doctoral Thesis. University of Claude Bernard, Lyon I.
0. Background to the human growth (auxiological) data.
1.1 Basic statistical and graphical analyses.
1.2 Application of the Preece Baines (PB) model to the study of human
1.3 Principal component analysis on longitudinal data and the PB model
2.1 STATIS applied to the description of the morphological development of
2.2 LONGI: theoretical exposition and an application to the description of
development of children.
Perrin, N.A., & Ashby, F.G. (1991).
A test for perceptual independence with dissimilarity data.
Applied Psychological Measurement, 15, 79-93.
Presents the dominance axiom as a test of dissimilarity data to determine
if the dimensions of
perceptual space are perceived independently, and as a diagnostic tool in
assessing the INDSCAL model's (Carroll and Chang, 1970)
assumption of independent dimensions. The general recognition theory of
similarity, which contains both the three-mode and INDSCAL multidimensional
scaling models as special cases, is used to motivate the test. General
recognition theory predicted consistent violations of the dominance axiom
with dependent dimensions, but not independent dimensions. A consistent
pattern of violations of dominance suggests that the three-mode model is
appropriate. When the test of dominance is satisfied, the INDSCAL model is
appropriate for the data.
Persat, H., & Chessel, D. (1989).
Typologie de distributions en classes de taille: intérêt dans
l'étude des populations de poissons et d'invertébrés
[Size-class ordination: A useful method in the
study of fish and invertebrate populations].
Acta Oecologica - Oecologia Generalis, 10, 175-195.
How best to perform a statistical analysis of a series of histograms is a
common question in population biology. This paper gathers a set methodological
proposals concerning Correspondence Analysis, in the prospect opened by
Badia & Do Chi (1976). Here we will present the functional representation
of factorial coordinates and eigenvalues, the smoothing of histograms by means
of the data resetting formula, and the projection of additional individuals
in the case of a spatio-temporal study. The Within-class Correspondece Analysis
(Benzecri, 1983) is introduced in the event of a multispecific structure.
These proposals make a coherent set of tools available in the field of population
biology. The examples concern the dynamics and the growth of invertebrate
populations (data from El Kallab-Wakim, 1978; and Moueza, 1975), and the
biology (disperal, growth, ecological patterns) of fish populations of the
French Upper Rhone River. Here the two approaches of Correspondence Analysis
as a method for processing contingency tables, and as a method of ordination
for ecological data tables are unified.
Pettersson, A. K., & Karlberg, B. (1997).
Simultaneous determination of orthophosphate and arsenate based on multi-way
spectroscopic-kinetic data evaluation.
Analytica Chimica Acta, 354, 241-248.
Comparison has been made of various multivariate calibration models for
the simultaneous determination of phosphate and arsenate. Both ions react with molybdate
and ascorbic acid forming a blue complex. Differences appear for the two ions both with
respect to spectral and kinetic behaviour. Multiway partial least squares (PLS) models
based on the combined effect of spectral and kinetic differences are superior to PLS
models based on either difference alone. The concentration working range was 0-88 mu
mol l(-1) for phosphate and 0-27 mu mol l(-1) for arsenate. The root mean square error
of prediction values were typically 6 (P) and 2 (As) mu mol l(-1) when calibration models
utilising 31 spectra for each solution were applied. The spectral range was 410-820 nm,
one absorbance value was measured every second nm (205 data points), and one scan/min was
performed. Phosphate catalyses the reaction between arsenate and the added reagents due
to the formation of one or more mixed complexes.
Pham, T.D., & Möcks, J. (1992).
Beyond principal component analysis: A trilinear decomposition model
and least squares estimation. Psychometrika, 57,
The paper derives sufficient conditions for the consistency and
asymptotic normality of the least squares estimator of a
trilinear decomposition model for multiway data.
Phillips, G.R., & Georghiou, S. (1993).
Global analysis of steady-state polarized
fluorescence spectra using trilinear curve
Biophysical Journal, 65, 918-926.
Global analysis using trilinear curve resolution
is described and shown to be a powerful method for
the resolution of polarized fluorescence data
arrays, in which the measured fluorescence
intensity is a separable function of polarization
orientation, excitation wavelength, and emission
wavelength. This methodology is applicable to
mixtures the components of which have linearly
independent excitation and emission spectra and
distinct anisotropies. Normalized excitation and
emission spectra of individual components can be
uniquely determined without prior assumptions
concerning spectral shapes (e.g., sum of
Gaussians) and without the uncertainties inherent
in bilinear techniques such as principal component
analysis or factor analysis. The normalized
excitation and emission vectors are combined with
the total absorption spectrum of the
multicomponent mixture to compute absolute
absorption and emission spectra. The precision of
this methodology is evaluated as a function of
noise, overlap, relative intensity, and anisotropy
difference between components using simulated
mixtures of the DNA bases. The ability of this
method to extract individual spectra from
steady-state fluorescence data arrays is
illustrated for mixtures containing two and three
Pieters, R.G.M., &
De Klerk-Warmerdam, M. (1996).
Ad-evoked feelings: Structure and impact on Aad
and recall. Journal of Business Research, 37,
This paper examines the structure of feelings that consumers experience
concurrently during exposure to print advertising and analyze how the structure affects
advertising processing and effectiveness. In study 1, a three-dimensional structure of the
experienced similarity of feelings is found: pleasantness, intensity and direction. In study 2,
three distinct bundles of feelings that consumers experience concurrently during exposure to
a set of print advertisements are recovered using three-mode principal components analysis:
unpleasant feelings, low-intensity pleasant feelings, and high-intensity pleasant feelings.
Unpleasant feelings and low-intensity pleasant feelings have a significant impact on attitude
to advertisement (Aad), high-intensity pleasant feelings have a significant impact
on advertising recall, and Aad and advertising recall are uncorrelated.
Pison, G., Van Aelst, S. & Willems, G. (2002).
Small sample corrections for LTS and MCD.
Metrika, 55, 111-123.
The least trimmed squares estimator and the minimum covariance determinant
estimator  are frequently used robust estimators of regression and of location and scatter,
Consistency factors can be computed for both methods to make the estimators consistent at the
normal model. However, for small data sets these factors do not make the estimator unbiased.
Based on simulation studies we therefore construct formulas which allow us to compute small
sample correction factors for all sample sizes and dimensions without having to carry out any
new simulations. We give some examples to illustrate the effect of the correction factor.
Pison, G., Rousseeuw, P. J., Filzmoser, P., & Croux, C. (2003).
Robust factor analysis.
Journal of Multivariate Analysis, 84, 145-172.
Our aim is to construct a factor analysis method that can resist the
effect of outliers. For this we start with a highly robust initial covariance estimator,
after which the factors can be obtained from maximum likelihood or from principal factor
analysis (PFA). We find that PFA based on the minimum covariance determinant scatter
matrix works well. We also derive the influence function of the PFA method based on
either the classical scatter matrix or a robust matrix. These results are applied to the
construction of a new type of empirical influence function (EIF), which is very effective
for detecting influential data. To facilitate the interpretation, we compute a cutoff
value for this EIF. Our findings are illustrated with several real data examples.
Pistonesi, M., Centurion, M. E., Band, B. S. F., Damiani, P. C., & Olivieri, A. C. (2004).
Simultaneous determination of levodopa and benserazide by stopped-flow injection
analysis and three-way multivariate calibration of kinetic-spectrophotometric data.
Journal of Pharmaceutical and Biomedical Analysis, 36, 541-547.
The simultaneous determination of levodopa and benserazide in
pharmaceutical formulations is described, based on the application of multidimensional
partial least-squares regression to the kinetic-spectrophotometric data provided by
diode-array detection within a stopped-flow injection method where analytes react with
periodate. Flow injection parameters were adequately optimized. Accurate analysis is
performed with no sample pre-treatment steps, and with minimum experimental effort.
Satisfactory recovery results were obtained on a number of synthetic and commercial
samples, in the latter case including the comparison with liquid chromatography
Pittam, J., Gallois, C., & Callan, V. (1990).
The long-term spectrum and perceived emotion. Speech
Communication, 9, 177-187.
This study shows systematic links between the long-term spectrum of voice
(LTS) and the
affective dimensions of control, arousal, and pleasure. Fifteen male and
15 female speakers of
three ethnic backgrounds - Australian, British, and Italian - recorded
three spoken passages.
A three-mode principal components analysis showed that the passages were
differentiated by the
LTS. In a second study using MANOVA, the three passages were then shown to
differently on the three affective dimensions. Finally, the LTS was shown
to be systematically
related to the affective dimensions in the ranges 0-350 Hz (control), 2-
2.5 kHz (arousal and
pleasure), and 4-10 kHz (control). No significant sex or ethnic group
Pittam, J., Gallois, C., Iwawaki, S.,
& Kroonenberg, P.M. (1995).
Australian and Japanese concepts of expressive behavior.
Journal of Cross-Cultural Psychology, 26, 451-473.
The present study explored the possibility of latent dimensionalities in
subjects' ratings of
expressive behaviours and examined the differences in these structures for
the two subject
groups. This was achieved using a form of three-mode principal component
in which components are computed for two of three modes.
Polissar, A.V., Hopke, P.K., Paatero, P., Kaufmann, Y.J., Hall, D.K.,
Bodhaine, B.A., Dutton, E.G., & Harris, J.M. (1999).
The aerosol at Barrow,
Alaska: long-term trends and source locations. Atmospheric Environment,
Aerosol data measured at Barrow, Alaska from 1977 to 1994 have been analyzed by
three-way positive matrix factorization (PMF3) by pooling all of the different
data into one large three-way array. The PMF3 analysis identified four factors
that indicate four different combinations of aerosol sources active throughout
the year in Alaska.
Pols, L.C. (1983).
Three-mode principal component analysis of confusion matrices, based on
the identification of Dutch consonants, under various conditions of noise
and reverberation.Speech Communication, 2, 275-293.
Dutch consonants, spoken in lists of two-syllable nonsense words
embedded in short carrier phrases, were identified by listeners
under 28 acoustic disturbance conditions. Results were summated
over the 6 26-40 yr old talkers and 5 18-23 yr old listeners. The
resulting three-dimensional stimulus configuration for the initial
consonants was stable and could be represented as a tetrahedron
with /z, s/, /m, n/, /p, t, k, b, d/, and /f, v, chi/ at the 4
corner points and /l, r, w, j, h/ in the center. The representation
was almost exclusively one-dimensional despite the 3 different
aspects (reverberation time, noise level, and noise consonant
identification; 26-40 yr old speakers & 18-23 yr old listeners)
Poppe, M., Croon, M., & Sluijtman, A. (1989).
De impliciete structuur van sociale waarden en reachtvaardigheidsregels in
een context van sociale relaties. [Implicit structure of social values and
justice rules in a context of social relations]. Nederlands Tijdschrift
voor de Psychologie, 44, 235-243.
Subject indicated how (un)usual each of 17 (role) relationships and 11
social values and justice rules was. A Tucker2 analysis of the data showed
a tw-dimensional implicit structure explaining about 65\% of the variance.
One dimension could be labelled as 'co-operation' and the other as
'competition'. The three justice rules fir well in this structure. Also the
implicit structure of scoial realtions could be interepreted in terms of
'co-operation' and 'competition'.
Porjesz, B., Almasy, L., Edenberg, H. J., Wang, K., Chorlian, D. B., Foroud, T.,
Goate, A., Rice, J. P., O'Connor, S. J., Rorhbaugh, J., Kuperman, S., Bauer, L. O.,
Crowe, R. R., Schuckit, M. A., Hesselbrock, V., Conneally, P. M., Tischfield, J. A.,
Li, T. K., Reich, T., & Begleiter, H.(2002).
Linkage disequilibrium between the beta frequency of the human EEG and a GABAa receptor gene
locus. Proceedings of the National Academy of Sciences, 99, 3729-3733.
Human brain oscillations represent important features of informa-tion
processing and are highly heritable. A common feature of beta
oscillations (13–28 Hz) is the critical involvement of networks of
inhibitory interneurons as pacemakers, gated by y-aminobutyric
acid type A (GABAA) action. Advances in molecular and statistical
genetics permit examination of quantitative traits such as the beta
frequency of the human electroencephalogram in conjunction with
DNA markers. We report a significant linkage and linkage disequi-librium
between beta frequency and a set of GABAa receptor
genes. Uncovering the genes influencing brain oscillations pro-vides
a better understanding of the neural function involved in
Povlsen, V. T., Rinnan, A., Van den Berg, F., ANdersen, H. J., amp; Thybo, A. K. (2003).
Direct decomposition of NMR relaxation profiles and prediction of sensory attributes
of potato samples.
Lebensmittel-Wissenschaft und -Technologie-Food Science and Technology, 36, 423-432.
In this paper the decomposition of low-field Carr-Purcel-Meiboom-Gill
(CPMG) NMR relaxation measurements on 23 raw potato categories was investigated. The
potato categories were formed from five different cultivars, each binned in 2 or 3 dry
matter intervals, sampled at two storage times. A novel data analytical tool-called
SLICING-revealed that different amounts of four distinct proton relaxation profiles could
describe the main variation in the data set. Magnitudes (scores) of the third and fourth
profile separated the potato cultivars, storage times, and dry matter content indicating
that properties related to fast relaxation times explain the differences between
cultivars and storage times for the potatoes. The concept of direct decomposition using
SLICING on low-resolution NMR data is a new approach in potato analysis and a promising
tool for obtaining more information about the structure and water distribution in food
Furthermore, the texture-related sensory attributes, hardness, cohesiveness, adhesiveness,
mealiness, graininess, and moistness of cooked potatoes were predicted by partial
least-squares regression (PLSR). Four different types of predictor variables derived from
the NMR relaxation curves were compared in the regression models: (i) the raw CPMG curves,
(ii) the parameters from the traditional bi-exponential fitting; (iii) the results from a
distribution analysis, and (iv) the scores from the SLICING model. The predictions based
on the distribution analysis performed worse than the first three procedures, which all
showed similar prediction ability. The advantage of the SLICING approach is in the
possibility to interpret physical properties, e.g. water distribution of the potato
Pravdova, V., Estienne, F., Walczak, B., & Massart, D.L. (2001).
A robust version of the Tucker3 model. Chemometrics and Intelligent Laboratory
Systems, 59, 75-88.
A new procedure for identification of outliers in Tucker3 model is proposed. It
is based on robust initialization of the Tucker3 algorithm using Multivariate trimming
or Minimum covariance determinant. The performance of the algorithm is tested
by a Monte Carlo study on simulated data sets and also on a real data set known
to contain outliers.
Pravdova, V., Boucon, C., de Jong, S. Walczak, B., & Massart, D. L. (2001).
Three-way principal component analysis applied to food analysis: an example.
Analytica Chimica Acta, 462, 133-148.
The purpose of the study is to showhowthe interpretation of a complex multivariate data array can significantly be improved
by the application of N-way principal component analysis (PCA). Two food related three-way data sets were studied; a sensory
and a chromatographic data array. The Parafac and the Tucker3 models were applied and results were compared. Both N-way
models presented here allow visualization of the data structure and give detailed information about the data set, notably
allowing to understand relationships between objects and variables.
Prazen, B. J., Bruckner, C. A., Synovec, R. E. & Kowalski, B. R.(1999).
Enhanced chemical analysis using parallel column gas chromatography with
single-detector time-of-flight mass spectrometry and chemometric analysis.
Analytical Chemistry, 71, 1093-1099.
A parallel gas chromatographic instrument with time-of-night mass spectrometric detection
(GC/TOF-MS) is reported. An injected sample is first split between two GC columns that provide
complementary separations. The effluent from the two columns is recombined prior to detection with
a single TOF-MS, Switching from single to parallel columns increases the chemical
selectivity of a GC/TOF-MS data set without increasing analysis time, by doubling the number
of peaks, or features, in the chromatographic dimension. The resulting analyzer can be used to
reduce analysis times for partially resolved peaks. Simulations compare the quantitative
precision of parallel-and single-column instruments using the generalized rank
annihilation method (GRAM), Results indicate that a parallel column GC/TOF-MS should substantially
improve the chemical selectivity and quantitative precision of the analysis relative to a
single-column instrument, For a column at half its peak capacity, for example, a single-column
instrument met the target precision less than 75% of the time, while a parallel-column instrument
achieved 95% success. Parallel-column analyses of methyl tert-butyl ether (MTBE) and benzene in
gasoline samples were also performed to support the simulation studies. An objective
chromatographic standardization technique corrected for retention time shifts before GRAM
was applied. Although MTBE and benzene were poorly resolved in the 40-s runs, chemometric
techniques successfully quantitated them.
Prazen, B. J., Johnson, K. J., Weber, A., & Synovec, R. E. (2001).
Two-dimensional gas chromatography and trilinear partial least squares for the quantitative analysis of aromatic and naphthene content in naphtha.
Analytical Chemistry, 73, 5677-5682.
Quantitative analysis of naphtha samples is demonstrated
using comprehensive two-dimensional gas chromatography (GC x GC) and
chemometrics. This work is aimed at providing a GC system for the
quantitative and qualitative analysis of complex process streams for
process monitoring and control. The high-speed GC x GC analysis of naphtha
is accomplished through short GC columns, high carrier gas velocities,
and partial chromatographic peak resolution followed by multivariate
quantitative analysis. Six min GC x GC separations are analyzed with
trilinear partial least squares (tri-PLS) to predict the aromatic and
naphthene (cycloalkanes) content of naphtha samples. The 6-min GC x GC
separation time is over 16 times faster than a single-GC-column standard
method in which a single-column separation resolves the aromatic and
naphthene compounds in naphtha and predicts the aromatic and naphthene
percent concentrations through addition of the resolved signals.
Acceptable quantitative precision is provided by GC x GC/tri-PLS..
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Algemene en Gezinspedagogiek - Datatheorie
Centre for Child and
Family Studies |
Department of Educational
The Three-Mode Company |
Education and Child Studies, Leiden University
Wassenaarseweg 52, 2333 AK Leiden, The Netherlands
Tel. *-31-71-5273446/5273434 (secr.); fax *-31-71-5273945
First version : 12/02/1997;