Three-Mode Abstracts, Part R

INDEX

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• Rajko, R. (2001).
  Calibration of chemical measurements. On quality of analytical information. Magyar Kemial Folyoirat, 107, 45-59.
  The author has been planning to publish a comprehensive review on calibration for a long time. However, it seems almost impossible task at all time. By all means it is necessary to look back, because it turned out that there are several emerging problems to investigate and to solve even in the simplest linear univariate cases. Looking ahead is also needed, because the difficult tensorial calibration models offer fruitful developing. The calibration is placed in the flow of the quality assurance of getting analytical information, and the calibration models are examined in compliance with it, The figures of merit are strongly important, they can serve as a guide for instrument design and characterization. Several questions have remained open for further scientific research. It is hoped some future results are due to the contribution of this paper.

• Ramaker, H.J., Van Sprang E.N.M., Gurden, S.P., Westerhuis J.A. & Smilde A.K. (2002)
  Improved monitoring of batch processes by incorporating external information Journal of Process Control, 12, 569-576
  In this paper an overview is given of statistical process monitoring with the emphasis on batch processes and the possible steps to take for improving this by incorporating external information. First, the general concept of statistical process monitoring of batches is explained. This concept has already been shown to be successful according to the number of references to industrial applications. The performance of statistical process monitoring of batch processes can be enhanced by incorporating external information. Two types of external information can be distinguished: batch-run specific and process specific information. Various examples of both types of external information are given. Several ideas of how to incorporate the external information in model development are discussed. The concept of incorporating process specific information is highlighted by an example of a grey model. This model is applied to a biochemical batch process that is spectroscopically monitored.

• Ramsay, J.O., Munhall, K.G., Gracco, V.L., & Ostry, D.J. (1996).
  Functional data analyses of lip motion. Journal of the Acoustical Society of America, 99, 3718- 3727.**
  The vocal tract's motion during speech is a complex patterning of the movement of many different articulators according to many different time functions. Understanding this myriad of gestures is important to a number of different disciplines including automatic speech recognition, speech and language pathologies, speech motor control, and experimental phonetics. Central issues are the accurate description of the shape of the vocal tract and determining how each articulator contributes to this shape. A problem facing all of these research areas is how to cope with the multivariate data from speech production experiments. In this paper techniques are described that provide useful tools for describing multivariate functional data such as the measurement of speech movements. The choice of data analysis procedures has been motivated by the need to partition the articulator movement in various ways: En effects separated from shape effects. Partitioning of syllable effects, and the splitting of variation within an articulator site from variation from between sites. The techniques of functional multivariate procedures such as analysis of variance and principal components analysis have their functional counterparts, and these reveal in a way more suited to the data the important sources of variation in lip motion. Finally, it is found that the analyses of acceleration were especially helpful in suggesting possible mechanisms. The focus is on using these speech production data to understand the basis principles of coordination. However, it is believed that the tools will have a more general use.

• Ramsay, J.O., Ten Berge, J., & Styan, G.P.H. (1984).
  Matrix correlation. Psychometrika, 49, 403-423.
  A correlational measure for an n by p matrix X and an n by q matrix Y assesses their relation without specifying either as a fixed target. This paper discusses a number of useful measures of correlation, with emphasis on measures which are invariant with respect to rotations or changes in singular values of either matrix. The maximization of matrix correlation with respect to tranformations XL and YM is discussed where one or both transformations are constrained to be orthogonal. Special attention is focussed on transformations which cause XL and YM to be n by s, where s may be any number between 1 and min(p, q). An efficient algorithm is described for maximizing the correlation between XL and YM where analytic solutions do not exist. A factor analytic example is presented illustrating the advantages of various coefficients and of varying the number of columns of the transformed matrices.

• Ray, W.J., & Cole, H.W. (1985).
  EEG alpha activity reflects attentional demands, and beta activity reflects emotional and cognitive process. Science, 228, 750-752.
  Two experiments were designed to examine the effects of attentional demands on the electroencephalogram during cognitive and emotional tasks. We found an interaction of task with hemisphere as ell as more overall parietal alpha for tasks not requiring attention to the environment, such as mental arithmetic, than for those requiring such attention. Differential hemispheric activation for beta was found most strongly in the temporal area for emotionally positive or negative tasks and in the parietal areas for cognitive tasks. These data were analyzed by analysis of variance (day x attentional focus x emotional valence x hemisphere) and by a factor analytic technique (PARAFAC).

• Ray, W.J., & Vasey, J. (1985).
  PARAFAC analysis of psychophysiological data. Psychophysiology, 22, 577.
  In this presentation Ray briefly discusses the underlying model in comparison to traditional factor analysis and illustrates the procedure with EEG data collected in the context of a study examining the processing of positive and negative emotion. The results of the PARAFAC analysis answer a variety of questions for the psychophysiologists. First, the analysis points out any outliers among the subjects. Second, the procedure shows how the EEG from the various sites of the cortex contribute to the processing of particular tasks. And third, the analysis can lend further support to the suggestion of the experimenter that a priori groupings of tasks (e.g., positive and negative emotional stimuli) are appropriately categorized.

• Rayens, W.S., & Mitchell, B.C. (1997).
  Two-factor degeneracies and a stabilization of Parafac. Chemometrics and Intelligent Laboratory Systems, 38, 173-181.
  Mitchell and Burdick (B.C. Mitchell, D.S. Burdick, An empirical comparison of resolution methods for three-way arrays, Chemom. Intell. Lab. Syst. 20 (1993) 149-161; B.C. Mitchell, D.S. Burdick, Slowly converging Parafac sequences: Swamps and two-factor degeneracies, J. Chemom. 8 (1994) 155-168.) uncovered an intriguing correspondence between the existence of Parafac swamps and the presence of two-factor degeneracies (2FDs). This observation, coupled with the recognition that post-swamp resolutions were generally better than pre-swamp resolutions allowed the user a method of detecting when a swamp had likely been encountered and, hence, when it was safe to assume that Parafac had converged to a suitable resolution. Still, this correspondence alone did not suggest how one might reduce the number of Parafac iterations required to move through a swamp. In this paper, it is noted that serious 2FDs must produce an identifiable ill-conditioning in the least squares estimation step in Parafac. Moreover, a serious 2FD is only one way this ill-conditioning may occur and, hence, it is more correct to say that swamps correspond to this ill-conditioning in general, rather than to the presence of 2FDs in particular. In an attempt to reduce the number of iterations that parafac spends in a swamp, a particular method of stabilization is employed and results are presented which suggest that the number iterations can often be greatly reduced.

• Rayner, J.C.W., Best, D.J., & Liddell, G.F. (1991).
  Optimal testing in a three way ANOVA model. Communications in Statistics - Simulation, 20, 411-425.
  Likelihood ratio, score and Wald tests are considered for a three-way random effects ANOVA model. Competitor tests are compared using criteria such as small sample power, asymptotic relative efficiency, and convenient null distribution. The final choice is between a new test and two tests long used in practice.

         

• Realo, A., Koido, K., Ceulemans, E., & Allik, J. (2002).
  Three components of individualism. European Journal of Personality, 16, 163-184.
  In this article, following an assumption that individualism and collectivism are separate factors, we have further established that three central components of individualism can be distinguished. In the first part of the article we examined whether the three proposed components of individualism—autonomy, mature self-responsibility, and uniqueness— can be distinguished from each other in one cultural context, Estonia. A new scale was developed to measure the three aspects of individualism which demonstrated both the reasonable internal-consistency reliability as well as convergent and divergent validity with several other measures of individualism and collectivism and related constructs. In the second part of the article we studied whether individualism generalizes across specific contexts or domains of social relationships, namely, across relations with family and close others; friends and peers; state and nation. The results of the three-mode principal component analysis showed that the individualistic tendencies of the respondents did not differ much while measured toward the three types of social relation.

• Redfield, J. (1978).
  TMFA: A FORTRAN program for three-mode factor-analysis and individual-differences in multidimensional-scaling. Educational and Psychological Measurement, 38, 793-795.
  Description of a program to perform all three methods of Tucker (1966) and also three-mode scaling (Tucker, 1972); includes fitting of the additive constant in case of input of similarities. Large number of options for input, analysis, and output.

  

• Redfield, J. & Stone, A. (1979).
  Individual view points of stressful life events. Journal of Consulting and Clinical Psychology, 47, 147- 154.
  85 students rated 44 stressful life events on six 21 interval bipolar scales. Analysis with T3. The events and scales factors were graphically rotated to simple structure. The 3 student factors by varimax. The factors were scaled according to Snyder & Wiggins (1970). Little numerical detail. Validation with outside variables.

• Reid, J.C. (1969).
  A three-mode factor analysis of students' perceptions of a university. The Journal of Experimental Education, 36, 93-96.
  The structure of students' responses on a semantic differential instrument to certain facets of their university was analyzes by Tucker's three-mode factor analytical procedure. Factors were identified and named for the R, P, Q and core matrices. Tucker's procedure offers the investigator additional information in interpreting such response data. The additional information obtained from Tucker's procedure over classical two-mode procedures, particularly the core box of interdependencies, was helpful in interpreting the semantic differences data.

  

• Reinikainen, S.-P., Laine, P., Minkkinen, P., & Paatero, P. (2001).
  Factor analytical study on water quality in Lake Saimaa, Finland. Fresenius Journal of Analytical Chemistry, 369, 727-732.
  Multivariate data analysis methods (4-way Candecomp-PARAFAC: model solved with Multilinear Engine (ME-1)) were used to interpret the data of over two decades to study the changes in the water of Lake Saimaa in Finland. Earlier studies have shown that it is difficult to extract the natural background from the other sources of variation. By using the multilinear model three interpretable factors representing natural and anthropogenic processes could be extracted. The natural long-term variation, seasonal fluctuation and dilution of discharges in the recipient area could be extracted into their own factors, which could be easily visualized. The variation could be also presented with estimated variation in the water quality parameters caused by each of these natural or anthropogenic processes.

• Reis, M.M., Gurden, S.P., Smilde, A.K., & Ferreira, M.M.C. (2000).
  Calibration and detailed analysis of second-order flow injection analysis data with rank overlap. Analytica Chimica Acta, 422, 21-36.
  With the current popularity of second-order (or hyphenated) instruments, there now exists a number of chemometric techniques for the so-called second-order calibration problem, i.e. that of quantifying an analyte of interest in the presence of one (or more) unknown interferent(s). Second-order instruments produce data of varying complexity, one particular phenomenon sometimes encountered being that of rank overlap (or rank deficiency), where the overall rank of the data is not equal to the sum of the ranks of the contributing species. The purpose of the present work is to evaluate the performance of two second-order calibration methods, a least squares-based and an eigenvalue-based solution, in terms of their quantitative ability and stability, as applied to flow injection analysis (FIA) data which exhibits rank overlap. In the presence of high collinearity in the data, the least squares methods is found to give a more stable solution. Two-mode component analysis (TMCA) is used to investigate the reasons for this difference in terms of the chemical properties of the species analysed. The success of second-order calibration of this data is found to depend strongly on the collinearity between the acidic and basic time profiles and the reproducibility of the pH gradient in the FIA channel, both of which are shown to be related to the pK_a values of the species.

• Reis, M. M., Biloti, D. N., Ferreira, M. M. C., Pessine, F. B. T., & Teixeira, G. M. (2001).
  PARAFAC for spectral curve resolution: A case study using total lumniscence in human dental tartar. Applied Spectroscopy, 5, 847-851.
  Chromophore identification in biological samples often requires the physical separation of the compounds, which can be difficult. Although there are several advantages to hyphenated spectroscopic techniques for identification of substances, complex mixtures of chromophores presenting overlapped spectra cannot be identified directly through this method. This work presents an application of chemometrics to compound identification in biological samples by a spectroscopic hyphenated technique using a curve resolution method. The PARAllel FACtor analysis model (PARAFAC), which has no rotational indeterminacy, was used for curve resolution of excitation-emission spectra of human dental tartars. PARAFAC was applied under constraints (i.e., unimodality and non-negativity) and evaluated with a validation procedure. The resolved profiles are porphyrinic-like spectra presenting excitation band maxima at 407, 416, and 431 nm in the Soret band region (390–440 nm) of these substances.

• Reis, M. M., & Ferreira, M. M. C. (1999).
  Curve resolution of simulated and total luminescence spectra by Generalized Rank Annihilation Method (GRAM). Quimica NOVA, 22, 11-17.
  A multivariate curve resolution method, "generalized rank annihilation method (gram)", is discussed and tested with simulated and experimental data. The analysis of simulated data provides general guidelines concerning the condition for uniqueness of a solution for a given problem. The second-order emission-excitation spectra of human and animal dental calculus deposits were used as an experimental data to estimate the performance of the above method. Three porphyrinic spectral profiles, for both human and cat, were obtained by the use of gram.

• Reis, M. M., & Ferreira, M. M. C. (2002b).
  PARAFAC with splines: a case study. Journal of Chemometrics, 16, 444-450.
  The PARAFAC model has been used in several applications in chemistry, e.g. for overlapped spectra resolution and second-order calibration. In general, the PARAFAC method uses a vector space approach by considering the matrices resulting from the decomposition as a collection of vectors. This paper presents a PARAFAC application where the factors resulting from the decomposition are considered as functions. The functional objects used for this are spline functions. The methodology used performs the Spline-PARAFAC decomposition based on the Bro-Sidiropoulos approach for the unimodality constraint. One of the advantages of using splines is the possibility of achieving a controlled degree of smoothing on the decomposed components. The amount of smoothing applied on the components in the presented methodology is controlled by a penalty parameter or by the number of basis functions. Thus Spline-PARAFAC requires the calculation of the parameter and the number of basis functions, which were determined in this work by using ordinary cross-validation (OCV). Spline-PARAFAC was applied to a carbon monoxide data set comprising concentrations measured every hour during the years 1997 and 1999 in the city of São Paulo, Brazil. Each data set was arranged in a three-way array of dimension (24 hours × 5 days × 52 weeks). Spline-PARAFAC showed a good performance, producing smoothed profiles describing the daily variations in emitted gas and the seasonal effects during the year.

• Reis, M. M., Ferreira, M. M. C. & Sarmento, S. B. S. (2002c).
  A multi-way analysis of starch cassava properties. Chemometrics and Intelligent Laboratory Systems, 64, 123-135.
  The original methods proposed by Ledyard R. Tucker during the 1960s present the rotational freedom problem, making the interpretation of their results rather difficult to be carried out. Aiming to make the multi-way data analysis more acceptable, this work suggests a methodology for extracting meaningful information from the data set. This methodology is based on the decomposition of data set in three-way blocks by using Tucker models. With the aim of keeping in one block similar information about the data properties, a decomposition based on a constrained Tucker model was used, where the core array has some of its elements fixed to zero. This methodology is successfully applied to a data set formed by physical and physicochemical properties of starches of four cassava cultivars, harvested at different ages during the period usually taken for harvest of industrial uses.

• Renaud, J.P., Davydov, D.R., Heirwegh, K.P.M., Mansuy, D., Hoa, G.H.B. (1996).
  Thermodynamic studies of substrate binding and spin transitions in human cytochrome p-450 3a4 expressed in yeast microsomes. Biochemical Journal, 319, 675-681.**
  An approach to the quantitative spectral analysis of substrate binding and inactivation of cytochrome p-450 in microsomes is described. The method is based on the application of the principal component analysis technique on the soret-region spectra measured at different temperatures at various concentrations of substrate. This approach allowed us to study the thermodynamic parameters of substrate binding and spin transitions in human cytochrome p-450 3a4 expressed in yeast (saccharomyces cerevisiae) microsomes. These parameters are discussed in comparison with the values reported earlier by ristau et al. [(1979) Acta Biol. Med. Ger. 38, 177-185] for rabbit liver cytochrome p-450 2b4 in solution with benzphetamine as a substrate. Our analysis shows the substrate-free states of 2b4 and 3a4 to be very similar. However, substrate binding seems to perturb haem-protein interactions in 3a4 in contrast with 2b4, where the effect of substrate binding on the thermodynamic parameters of spin transitions was insignificant. The implication of the results for the mechanism of substrate-induced spin shift is discussed.

• Reyment, R.A. (1997).
  Multiple group principal component analysis. Mathematical Geology, 29, 1-16.
  Common Principal Component Analysis is a generalization of standard principal components to several groups under the rigid mathematical assumption of equality of all latent vectors across groups (i.e., principal component directions), whereas the latent roots are allowed to vary between groups (differing inflations of dispersion ellipsoids). In practice, data that fulfill these strict requirements are relatively rare. Examples from palaeontology are used to illustrate the principles. Compositional data can be made to fit the Common Principal component (CPC) model by the appropriate logratio covariance matrix.

• Reyment, R.A., & Naidin, D.P. (1962).
  Biometric study of actinocamax verus s.1. from the upper cretaceous of the Russian platform. Contributions in Geology. Acta Universitatis Stockholmiensis, 9, 147-206.
  This paper studies belemnite species from 9 localities. Primarily interesting because it contains a set of correlation matrices which were analysed later with three-mode methods (see Reyment, 1997).

  
         

• Richards, E., Bessant, C., & Saini, S. (2003).
  Multivariate data analysis in electroanalytical chemistry. Electroanalysis, 14, 1533-1542.
  Data analysis is becoming an increasingly important aspect of electroanalytical chemistry, as voltammetric techniques and electrode arrays become ever more popular as diagnostic tools. Modern data analysis techniques promise to help us make full use of the large amounts of data collected, allowing electroanalytical chemists to get more out of their existing instruments, and paving the way for new measurement approaches. Ibis article provides an overview of the most widely used multivariate techniques in electroanalysis, citing specific examples of how they have been applied, and looking at their relative merits. As in other areas of analytical science, no single technique is applicable to all applications and the running of controls and appreciation of the applications and limitations of each technique is essential.

• Rincon, F., Johnson, B., Crossa, J., & Taba, S. (1997).
  Identifying subsets of maize accessions by three-mode principal component analysis. Crop Science, 37, 1936-1942.
  Genebank accessions are a potential source of genetic variability for maize (Zea mays L.) breeding programs. Identification of useful individual entries is commonly based on the expression of one or more attributes in different sites or environments. This study was motivated by the need to identify a subset of Caribbean accessions for introgression into elite germplasm of the western Corn Belt. Thus the objective,vas to identify potentially useful Caribbean accessions, based on (i) simultaneous consideration of six agronomic attributes deemed economically important for the western Corn Belt, and (ii) response patterns observed across four sharply contrasting environments. Both (i) and (ii) were addressed by means of three-mode principal component analysis (PCA) of data on six agronomic and morphological attributes for 184 Caribbean maize accessions evaluated at four environments. Three-way data were analyzed by three-mode PCA, and based on (i) and (ii), two joint plots were generated. From the PCA and joint plots, two subsets of accessions were identified. First, a subset of 14 accessions having good yield, intermediate plant height, and average days to anthesis was identified. Secondly, a subset of 10 accessions having average performance over all environments was identified. Two accessions were common to both subsets. Jointly, the two approaches produced a combined subset of 22 accessions, representing 12% of the total evaluated, and including representatives of 11 maize races. Three-mode PCA integrated information on accessions, attributes, and environments, and provided a means of simultaneously visualizing these three types of information. Three-mode PCA can complement standard methodologies used by plant breeders for identification of potentially useful accessions in introgression programs.

• Rikken, F., Kiers, H.A.L., & Vos, R. (1995).
  Mapping the dynamics of adverse drug reactions in subsequent time periods using INDSCAL. Scientometrics, 33, 367-380.
  In this study we have focused on the problem of mapping the dynamics of co-word-matrices from subsequent time periods. Methods for mapping dynamics are important for following trends in research. We have explored the possibilities of a three way multidimensional scaling method, INDSCAL. We are especially interested to find relations between adverse drug reactions and other words in co-word-matrices from a medical field. Second we want to explore whether the relations between adverse drug reactions and other words have changed in subsequent time periods. The results show that INDSCAL can be a useful tool for mapping dynamics.

• Riu, J., & Bro, R. (2003).
  Jack-knife technique for outlier detection and estimation of standard errors in PARAFAC models Chemometrics and Intelligent Laboratory Systems, 65, 35-49.
  In the last years, multi-way analysis has become increasingly important because it has proved to be a valuable tool, e.g. in interpreting data provided by instrumental methods that describe the multivariate and complex reality of a given problem. Parallel factor analysis (PARAFAC) is one of the most widely used multi-way models. Despite its usefulness in many applications, up to date there is no available tool in the literature to estimate the standard errors associated with the parameter estimates. In this study, we apply the so-called jack-knife technique to PARAFAC in order to find the associated standard errors to the parameter estimates from the PARAFAC model. The jack-knife technique is also shown to be useful for detecting outliers. An example of fluorescence data (emission/excitation landscapes) is used to show the applicability of the method. (C) 2002 Elsevier Science B.V. All rights reserved.

• Rizzi, A. (1988).
  On the average matrix. In H.H. Bock (Ed.), Classification and related methods of data analysis (pp. 533-540).
  Amsterdam: Elsevier.
  This paper proposes to construct a mean matrix of a set of matrices such that the standard deviations for each averaged variable is equal to the mean standard deviation.

• Rizzi, A. (1989a).
  On the principal component analysis of three-way data matrices. In Y. Dodge (Ed.), Statistical Data Analysis and Inference, 501- 516.
  This paper is concerned with the Principal Component Analysis of three-way data matrices. Principal matrices are a linear combination of K matrices observed on K different occasions or sources of data; they have maximum Euclidean norm and they are incorrelated. After some remarks on centering three-way data matrices, the author introduces, also, some new definitions for the standardization of multiple matrices. Applications are found at the end of the paper.

• Rizzi, A. (1989b).
  On the synthesis of three way data matrices. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 143-154).
  Amsterdam: Elsevier.
  In this paper a new type of matrix is proposed which has the property that the deviance of the variables is equal to the mean deviance. Also several principal matrices are proposed and their analysis is discussed.

• Rizzi, A. (1989c).
  Clustering per le matrici a tre vie [Clustering for three-way matrices]. Statistica, 49, 195-208.
  Cluster analysis for three-way and three-mode data matrices can be done in various ways, in particular any of the three modes may be clustered, or the clustering can involve more than one mode. In this paper, after reviewing the INDCLUS and ADCLUS algorithms, two more recent proposals by Bellacicco & Ronzoni (1988) and Chiodi (1989) are illustrated. In addition a new algorithm, PRINCLUS, based on principal matrices is presented.

• Rizzi, A., & Vichi, M. (1992).
  Relations between sets of variates of a three-way data set. Statistica Applicata, 4, 635-651.
  In this paper we analyze two types of relations between sets of variates: the correlation and the connection (or dependence). Their measures can be derived from a general index that is examined. Furthermore, we discuss new results on Principal Matrices Analysis (PMA) and Factorial Matrices Analysis (FAMA), two techniques based on the relations between sets of variates of a three-way data matrix due to the sharing of common sets of factors.

  

• Rizzi, A., & Vichi, M. (1995a).
  Representation, synthesis, variability and data preprocessing of a three-way data set. Computational Statistics & Data Analysis, 19, 203-222.
  In the first section of this paper the structures (vectors and matrices) on which three-way data set X can be organized is described, as well as the information that can be pointed out when using these structures. Many three-way analyses are based on pooled representations of X, that are systematically studied. The information given by a three-way data set can be synthesized according to the structures utilized to represent X. In the second section one-way, two-way and three-way syntheses of X are defined. Also the variability of a three-way data set is evaluated, in section three, according to three different levels: one-way or fiber variability, two-way or slab variability and three-way variability. The syntheses and the variability indices of X can be used for data preprocessing of X, which is here discussed in section four. Furthermore, section five discusses the Principal Matrices Analysis on the base of three-way variability indices.

  
             

• Robert, P., & Escoufier, Y. (1976).
  A unifying tool for linear multivariate statistical methods: The RV-coefficient. Applied Statistics, 25, 257-265.
  Consider two data matrices on the same sample of n individuals, X(p × n), Y(q × n). From these matrices, geometrical representations of the sample are obtained as two configurations of n points, in domain R^p and domain R^q. It is shown that the RV-coefficient (Escouffier, 1970, 1973) can be used as a measure of similarity of the two configurations, taking into account the possibly distinct metrics to be used on them to measure the distances between points. The purpose of this paper is to show that most classical methods of linear multivariate statistical analysis can be interpreted as the search for optimal linear transformations or, equivalently, the search for optimal metrics to apply on two data matrices on the same sample; the optimality is defined in terms of the similarity of the corresponding configurations of points, which, in turn, calls for the maximization of the associated RV-coefficient. The methods studied are principal components, principal components of instrumental variables, multivariate regression, canonical variables, discriminant analysis; they are differentiated by the possible relationships existing between the two data matrices involved and by additional constraints under which the maximum of RV is to be obtained. It is also shown that the RV-coefficient can be used as a measure of goodness of a solution to the problem of discarding variables.

• Robert, P., Bertrand, D., & Sire, A. (1992).
  Identification of chemical-constituents by multivariate near-infrared spectral imaging. Analytical Chemistry, 64, 664-667.
  A near-infrared imaging spectroscopic system was tested to identify three main components of wheat (bran, gluten, starch). The system described below permitted the recording of images between 900 and 1900 nm by steps of 50 nm. A direct examination of images showed that starch was correctly identified at 1550 nm. However, such a direct study of images was not sufficient to characterize all the constituents. The images, therefore, were linearly combined by applying discriminant analyses. The more revelant wavelengths (950, 1450, 1500 nm) were determined using a stepwise discriminant analysis, and a mapping of the chemical constituents was obtained by applying a canonical discriminant analysis. In the segmented image, the percentages of well-classified pixels were 92 for bran, 95 for gluten and 99 for starch.

• Rocci, R. (1992a).
  Scalar product and synthesis of s-matrices. Statistica Applicata, 4, 693-699.
  In this paper an alternative view is given of the Euclidean scalar product between symmetric positive semidefinite matrices, characterizing a matrix on the grounds of its spectral decomposition. Following this approach we reconsider the "compromise matrix" and "mean matrix" methods tacking into account the rank of the "compromise" or "mean" matrix.

• Rocci, R., & Ten Berge, J.M.F. (1994).
  A simplification of a result by Zellini on the maximal rank of symmetric three-way arrays. Psychometrika, 59, 377-380.
  Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of order n × n as a linear combination of 1/2n(n + 1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetric n × n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In this paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n × n × p array consisting of p symmetric n × n slabs has maximal rank 1/2n(n + 1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n + 1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n - 1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large as p, the number of matrices analyzed.

• Rocci, R., & Ten Berge, J. M. F. (2002).
  Transforming three-way arrays to maximal simplicity Psychometrika, 67, 351-365.
  Transforming the core array in Tucker three-way component analysis to simplicity is an intriguing way of revealing structures in between standard Tucker three-way PCA, where the core array is unconstrained, and CANDECOMP/PARAFAC, where the core array has a generalized diagonal form. For certain classes of arrays, transformations to simplicity, that is, transformations that produce a large number of zeros, can be obtained explicitly by solving sets of linear equations. The present paper extends these results. First, a method is offered to simplify J x J x 2 arrays. Next, it is shown that the transformation that simplifies an I x J x K array can be used to also simplify the (complementary) arrays of order (J K - I) x J x K, of order I x (I K - J) x K and of order I x J x (I J - K). Finally, the question of what constitutes the maximal simplicity for arrays (the maximal number of zero elements) will be considered. It is shown that cases of extreme simplicity, considered in the past, are, in fact, cases of maximal simplicity.

• Rodriguez-Cuesta, M. J., Boque, R., Rius, F. X., Zamora, D. P., Galera, M. M., & Frenich, A. G. (2003).
  Determination of carbendazim, fuberidazole and thiabendazole by three-dimensional excitation-emission matrix fluorescence and parallel factor analysis. Analytica Chimica Acta, 491, 47-56.
  We simultaneously determined carbendazim, fuberidazole and thiabendazole by excitation-emission matrix (EEM) fluorescence in combination with parallel factor analysis (PARAFAC). Three-way deconvolution provided the pure analyte spectra from which we estimated the selectivity and sensitivity of the pesticides, and the relative concentration in the mixtures from which we established a linear calibration. Special attention was given calculating such figures of merit as precision, sensitivity and limit of detection (LOD), derived from the univariate calibration curve. The method, which had a relative precision of around 2-3% for the three pesticides, provided limits of detection of 20 ng ml(-1) for carbendazim, 4.7 ng ml(-1) for thiabendazole and 0.15 ng ml(-1) for fuberidazole. The accuracy of the method, evaluated through the root mean square error of prediction (RMSEP), was 27.5, 1.4, and 0.03 ng ml(-1), respectively, for each of the pesticides.

• Röhmel, J., Streitberg, B., & Herrmann, W. (1983).
  The COMSTAT algorithm for multimodal factor analysis: An improvement of Tucker's three-mode factor analysis method. Neuropsychobiology, 10, 157-163.
  Three-mode factor analysis, as developed by Tucker (1966), is a method for nonredundant representation of data arrays with three subscripts. Tucker, however, did not succeed in obtaining a least-squares solution for his model. In this paper a necessary condition is derived for a least-squares solution and an algorithm is constructed which improves any given initial solution in the least-squares sense. Shown is that this algorithm converges to a representation satisfying the necessary conditions for a least-squares solution. The method is in no way restricted to three dimensions, but can be applied to any multimodel data array.

• Rösler, F. (1972).
  Dimensionen der Aktivitäten und deren Beziehungen zu den Persönlichkeitsfaktoren "Extraversion/Introversion" und "Neurotizismus" sensu Eysenk. Unpublished master thesis, University of Hamburg, Hamburg, FRG.
  (See Rösler, 1975.)

• Rösler, F. (1975).
  Die Abhängigkeit des Elektroenzephalogramms von den Persönlichkeitsdimensionen E und N sensu Eysenk und unterschiedlich aktivierenden Situationen. Zeitschrift für experimentelle und angewandte Psychologie, 22, 630-667.
  As part of a larger study a reduction of frequency spectra was obtained via T3 on the data of 32 subjects, 18 measuring periods, and 40 frequencies. However, only principal components was performed on the frequencies mode. The factors were scaled as suggested by Bartussek (1973).

  

• Rösler, F. (1979).
  Identifying interindividual judgment differences: INDSCAL or three-mode factor analysis. Multivariate Behavioral Research, 14, 145-167.
  Seventy subjects judged 9 German key-politicians on 22 bipolar 7-point attribute scales. The proximity matrices from the computed profile similarities were analyzed with INDSCAL. T3 was applied to the original data. The results suggested that T3 was more appropriate to describe the interindividual judgement differences when correlated with external variables.

• Rösler, F., Jesse, J., Manzey, D., & Grau, U. (1982).
  Ist das LM-Gitter nur ein LM-Test? Eine dreimodale Faktorenanalyse des LM-Gitters für Kinder (Schmalt). (Is the LM-Gitter only an LM test? A trimodal factor analysis of the LM-Gitter für Kinder (Schmalt).) Diagnostica, 28, 131-145.
  Three-mode factor analysis was used to check Need Achievement Scale for Children for systematic situation-specific effects. Data of 104 boys and 94 girls were analyzed. Factorization of the statement mode resulted in a factor structure that separated the 3 subscales of the test in the same manner as reported by Schmalt. Factorization of the situation mode reveals a Woodrow structure only, thus pointing to time-correlated changes in response behavior and not to differences related to the motivational quality of the situations.

  

• Ross, R.T., Lee, C.-H., Davis, C.M., Ezzeddine, B.M., Fayyad, E.A., & Leurgans, S.E. (1991).
  Resolution of the fluorescence spectra of plant pigment-complexes using trilinear models. Biochimica et Biophysica Acta, 1056, 317-320.
  The intensity of fluorescence from a pigment is separately linear in functions of excitation wavelength, emission wavelength, and any chemical treatment which alters overall fluorescence yield. This multiple linearity permits the use of an extension of principal components analysis to resolve overlapping spectra without the use of any additional information. The method is used to resolve the spectra of pigment complexes in pea thylakoids, using the concentration of Mg²⁺ as the chemical treatment variable. Two components could be resolved accurately. One has little or no dependence on Mg²⁺; the other, with an excitation spectrum resembling LHC II, has a dependence on Mg²⁺ which follows the hill equation with a binding constant of 0.4-0.6 mM and a hill coefficient of 2.4-3.1.

• Ross, R.T., & Leurgans, S.E. (1995).
  Component resolution using multilinear models. In K. Sauer (Ed.), Biochemical spectroscopy (pp. 679-700).
  San Diego: Academic Press. [Methods in Enzymology, 246].
  In many circumstances, the spectroscopic properties of a biological specimen are a composite of the properties of several different chromophores within the specimen, and the investigator would like to know the properties of the individual chromophores. Physical separation of the chromophores is often difficult. Thus, one looks to mathematical methods for separation of the contributions of the different chromophores. This chapter focuses on the application of multilinear models. A major advantage of this class of methods is that component resolution may often be achieved with no prior information about the properties of the components. Bilinear models can be applied to many kinds of spectroscopy, and they have been widely used. In this chapter the focus is on the application of trilinear models in fluorescence spectroscopy. However, these trilinear models are also applicable to other kinds of excited-state spectroscopy, such as transient absorption spectroscopy. Along the way, bilinear and other models, including global analysis, are also discussed.

• Rowe, H.A.H. (1979).
  Three-mode factor analysis: Problems of interpretation and possible solutions. Australian Psychologist, 14, 222-223 (abstract of paper presented at the 14th Annual Conference of the Australian Psychological Society, University of Tasmania, 30. August, 1979).
  A summary of applications with T3 is presented. An example (18 problem solving strategies on 12 tasks of varying content and difficulty by 89 subjects) is discussed on a conceptual level; no numerical details are given. Special attention is given to meaningful interpretations of the core matrix in relation to different types of standardizations of the raw data. Reification with external variables.

  
             

• Ruch, W. (1980).
  Gemeinsame Strukturen in Weizbeurteilung und Persönlichkeit. Versuch einer empirischen Integration des Gegenstandsbereiches Witzbeurteilung in die differentielle Psychologie. Unpublished doctoral thesis, University of Graz, Austria.
  110 subjects rated 48 jokes on 5 seven-point scales. T3 using the scaling of Bartussek (1973). The subject factors were in- terpreted using external variables. Standardization over all jokes-scales combinations. Varimax rotation for jokes, promax for scales. Detailed analysis of subjects and core matrix using 'ideal subjects' and the extended core matrix.

  

• Ruch, W. (1981).
  Witzbeurteilung und Persönlichkeit: Ein trimodale analyse. Zeitschrift für Differentielle und Diagnostische Psychologie, 2, 253-273.
  110 subjects rated 48 jokes on 5 seven-point scales. T3 using the scaling of Bartussek (1973). The subject factors were in- terpreted using external variables. Standardization over all jokes-scales combinations. Varimax rotation for jokes, promax for scales. Detailed analysis of subjects and core matrix using 'ideal subjects' and the extended core matrix.

• Ruckebusch, C., Duponchel, L., Sombret, B., Huvenne, J. P., & Saurina, J. (2003).
  Time-resolved step-scan FT-IR spectroscopy: Focus on multivariate curve resolution. Journal of Chemical Information and Computer Sciences, 43, 1966-1973.
  The present paper describes the application of step-scan FT-IR spectroscopy in combination with chemometric analysis of the spectral data for the study of the photocycle of bacteriorhodopsin. The focus is on the performance of this instrumentation for time-resolved experiments. Three-dimensional data-spectra recorded over time-are studied using various factor analysis techniques, e.g., singular values decomposition, evolving factor analysis, and multivariate curve resolution based on alternating least squares. Transient intermediates formed in the time domain ranging from 1 mus to 6.6 ms are clearly detected through reliable pure time evolving profiles. At the same time, pure difference absorbance spectra are provided. As a result, valuable information about transitions and dynamics of the protein can be extracted. We conclude first that step-scan FT-IR spectroscopy is a useful technique for the direct study of difficult photochemical systems. Second, and this is the essential motivation of this paper, chemometrics provide a step forward in the description of the photointermediates.

• Ruckebusch, C., Duponchel, L., Huvenne, J. P., & Saurina, J. (2004).
  Multivariate curve resolution of step-scan FTIR spectral data. Vibrational Spectroscopy, 35, 21-26.
  Step-scan FTIR is a time-resolved spectroscopic technique. It can successfully probe transient photochemical systems through two-way data sets: spectra are recorded as the system is evolving with time. This work focuses on chemometrics for time-resolved data sets and in particular on the potential of the multivariate curve resolution (MCR) based on alternating least squares. The usefulness of this approach is demonstrated in a case study, which involves several intermediate states with time constant ranging from the microsecond to the millisecond. The chemometrics methodology is detailed in two points. MCR relies on the bilinear decomposition of an evolving multi-component data set into the product of two matrices related respectively to the time and to the spectral domains of the extracted components. The first aspect concerns the determination of the number of components observable in the spectral data set and the calculation of initial estimates of either the time or spectral profiles. Principal component analysis based methods such as evolving factor analysis are performed and finally allow a first insight into the contributions of the four intermediates extracted. The second point concerns the alternating least squares optimization under natural constraints to recover pure intensity profiles and pure difference spectra profiles. The characterization of each photo-intermediate unaccessible individually except in non-physiological conditions is thus assessed without any assumption about the kinetic of the system.

• Russell, D. (1982).
  The Causal Dimension Scale: A measure of how individuals perceive causes. Journal of Personality and Social Psychology, 42, 1137-1145.
  The studies reported describe the development of the Causal Dimension Scale, a measure designed to assess how the attributor perceives the causes he or she has stated for an event. A three-mode factor analysis confirmed the three-dimensional structure of the scale.

  
         

• Rychlak, J.F., Flynn, E.J. & Burger, G. (1979).
  Affection and evaluation as logical processes of meaningfulness independent of associative frequency. Journal of General Psychology, 100, 143-157.
  Forty-three undergraduates rated 25 CVC-trigrams and 25 words and paralogs according to 7 different instructions. The trigrams and the words-paralogs were analysed separately with T3. Only the instructions were interpreted in both cases, as only one factor was present in the verbal materials and the subject matrix suggested the presence of only one factor. Because of the decision to retain only one factor for subjects and word materials, it was decided not to carry out a 'core matrix analysis.

   

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