Three-Mode Abstracts, Part P

INDEX

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• 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, 38, 223-242.
  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 model. Journal of Computational and Graphical Statistics, 8, 854- 888.
  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 three-way algorithm. 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, 60, 25-41.
  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. Unpublished manuscript.
  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 are "large". 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, 14, 241-259.
  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 relations.

• Papagni, E. (1992).
  High-technology exports of EEC countries - persistence and diversity of specialization patterns. 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 O_k operators associated with the occasions, in interstructure-compromise-intrastructure methods is investigated. Particularly, in the case of K matrices X_k units per variables, both Escoufier operators W_k and covariance matrices R_k can be used. A comparison of the solutions obtained in the two cases O_k=W_k and O_k=R_k is proposed using some proximity indices between interstructures, between units representations and between variables representations. An illustrative example is presented concerning the economic-financial 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 further analysis.

• Pedersen, D. K., Munck, L. & Engelsen, S. B. (2002).
  Screening for dioxin contamination in fish oil by PARAFAC and N-PLSR analysis of fluorescence landscapes. 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 techniques. 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 growth.
  1.3 Principal component analysis on longitudinal data and the PB model parameters.
  2.1 STATIS applied to the description of the morphological development of children.
  2.2 LONGI: theoretical exposition and an application to the description of the morphological 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 most 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, 203-215.
  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 resolution. 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 components.

  
   

• Pieters, R.G.M., & De Klerk-Warmerdam, M. (1996).
  Ad-evoked feelings: Structure and impact on A_ad and recall. Journal of Business Research, 37, 105-114.
  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 (A_ad), high-intensity pleasant feelings have a significant impact on advertising recall, and A_ad 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 [6] 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 measurements.

• 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 be perceived 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 effects were found.

• 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 analysis (Tuckals2) 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, 33, 2441-2458.
  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 information processing.

• 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 products. 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 samples.

  
       

• 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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