Three-Mode Abstracts, Part M

INDEX

• Ma, B., Gemperline, P. J., Cash, E., Bosserman, M., & Comas, E. (2003).
  Characterizing batch reactions with in situ spectroscopic measurements, calorimetry and dynamic modeling. Journal of Chemometrics, 17, 470-479.
  A method for fully characterizing consecutive batch reactions using self-modeling curve resolution of in situ spectroscopic measurements and reaction energy profiles is reported. Simultaneous measurement of reaction temperature, reactor jacket temperature, reactor heater power and UV/visible spectra was made with a laboratory (50 ml capacity) batch reactor equipped with a UV/visible spectrometer and a fiber optic attenuated total reflectance (ATR) probe. Composition profiles and pure component spectra of reactants and products were estimated without the aid of reference measurements or standards from the in situ U-V/visible spectra using non-negative alternating least squares (ALS), a type of self-modeling curve resolution (SMCR). Multiway SMCR analysis of consecutive batches permitted standardless comparisons of consecutive batches to determine which batch produced more or less product and which batch proceeded faster or slower. Dynamic modeling of batch energy profiles permitted mathematical resolution of the reaction dose heat and reaction heat. Kinetic fitting of the in situ reaction spectra was used to determine reaction rate constants. These three complementary approaches permitted simple and rapid characterization of the reaction's rate of reaction, energy balance and mass balance.

• MacCallum, R.C. (1974).
  A comparison of two individual differences models for multidimensional scaling: Carrol and Chang's INDSCAL and Tucker's three-mode factor analysis. Unpublished doctoral thesis, University of Illinois, Urbana, Illinois. (Dissertation Abstracts International, 1975, 35 (7-B), 3619).
  (See MacCallum, 1976a, 1976b.)

• MacCallum, R.C. (1974b).
  Relations between factor analysis and multidimensional scaling. Psychogical Bulletin, 81, 505-516.
  Treats the differences between factor analysis and scaling in general. Discusses shortly the relation between INDSCAL and Tucker's three-mode scaling.

• MacCallum, R.C. (1976).
  Effects on INDSCAL of non-orthogonal perceptions of object space dimensions. Psychometrika, 41, 177-188.
  Investigation into the effect of violations of assumptions of the INDSCAL model. Relation between INDSCAL and Tucker (1972) model is discussed; the latter is recommended when violation of assumptions in the former are suspected.

  

• MacCallum, R.C. (1976b).
  Transformation of a three-mode multidimensional scaling solution to INDSCAL form. Psychometrika, 41, 385-400.
  Relations between Tucker (1972) and INDSCAL are discussed. A gradient procedure is developed which seeks to diagonalize the core matrix in all its frontal planes. A necessary condition for the extended core matrix to be diagonal is that the core matrix is diagonal. The developed procedure can be seen as a first step towards checking the assumptions of the INDSCAL model, as the latter is defined to have a diagonal extended core matrix. It is also shown that a core matrix consisting of two frontal planes can always be diagonalised. The transformation procedure is applied to the Tucker (1972) data, and to Jones & Young data consisting of judgments of 19 respondents about the similarities of 17 stimulus persons (see also De Leeuw & Pruzansky, 1978).

• MacCallum, R.C. (1978).
  TRINFORM: A computer program to transform a three-mode multidimensional scaling solution to INDSCAL form. Applied Psychological Measurement, 2, 62.
  TRINFORM is a program which transforms a three-mode multidimensional scaling solution (Tucker, 1972) to the form, as nearly as possible, of an INDSCAL solution (Carroll and Chang, 1970). Using a method suggested by MacCallum (1976), this is achieved by transforming the three-mode core matrix, a matrix consisting of as many sections as person dimensions with each section specifying how each person dimension is related to perception of the stimulus dimensions.

• MacGregor, J.F., & Kourti, T. (1995).
  Statistical process control of multivariate processes. Control Engineering Practice, 3, 403-414.
  With process computers routinely collecting measurements on large numbers of process variables, multivariate statistical methods for the analysis, monitoring and diagnosis of process operating performance have received increasing attention. Extensions of traditional univariate Shewhart, CUSUM and EWMA control charts to multivariate quality control situations are based on Hotelling's T² statistic. Recent approaches to multivariate statistical process control which utilize not only product quality data (Y), but also all of the available process variable data (X) are based on multivariate statistical projection methods (Principal Component Analysis (PCA) and partial least squares (PLS)). This paper gives an overview of these methods, and their use for the statistical process control of both continuous and batch multivariate processes. Examples are provided of their use for analysing the operations of a mineral processing plant, for on-line monitoring and fault diagnosis of a continuous polymerization process and for the on-line monitoring of an industrial batch polymerization reactor.

• Maeda, S. (1992).
  Articulatory modeling of the vocal-tract. Journal de physique iv, 2, 307-314.**
  Recently, there is some renewed interest in the representation of speech at the articulatory level. For such a representation, it is essential to have a parsimonious model which is capable of describing accurately and meaningfully the states of the vocal tract during speech production. In this paper, first we briefly review various types of models proposed in the past. Then we focus our attention to a specific type of articulatory models, which are derived from a factor analysis of sagittal x-ray film data. The advantage of this type of model is that the vocal tract profiles are specified by a small number of the model parameters, less than 10, and their values can be directly calculated, not estimated, from data. Second, we describe the particular relationships between the model parameters and acoustics parameters (formant frequencies) that suggest acoustic compensation between different articulators as the lower jaw and tongue-body. Our analysis of x-ray data shows that such compensation is actually used in speech production, and that the larger variability in articulatory "targets" than in acoustic ones is, in part, due to compensation.

• Maeder, M., Neuhold, Y. M., Olsen, A., Puxty, G., Dyson, R., & Zilian, A. (2002).
  Rank annihilation correction for the amendment of instrumental inconsistencies. Analytica Chimica ActaIV, 464, 249-259.
  The globalisation of the analysis of a series of individual measurements often results in more robust and reliable outcomes. However, instrumental drifts that can occur between individual measurements destroy the ideal data structure and thus the advantages. A method based on rank annihilation factor analysis (RAFA) is introduced for the correction of several types of instrumental inconsistencies. It can be applied to many series of bilinear datasets. Experimental examples discussed in this paper comprise the successful correction of non-uniform retention time drifts in chromatography due to temperature or pressure changes, wavelength shifts in IR spectroscopy in an industrial control situation, and background absorption shifts in UV-VIS spectroscopy applied to equilibrium investigations.

• Mager, P. P. (1996).
  A random number experiment to simulate resample model evaluations. Journal of Chemometrics, 10, 221-240.
  Gaussian distributed scores lying within the range from +4 to -4 were calculated using a random generator. The random sample was divided sequentially into three subsamples with equal and unequal sizes. This classification leads to different internal correlation-regression structures depending on the subsample size. The subsamples show also departures from multivariate normality. This is misleading for hypothesis testing. Randomness of raw and vector-valued observations is unpredictable after subsampling. Furthermore, the chance to get outlying observations must be taken into account. Two sets of variables drawn from the subsamples did not show remarkable relationships. The situation changed after omission of certain variables and subsample members. The formally significant equations are artifacts and show the real danger of a selection of regressors extracted from a variable pool of individual subsamples. Consequently, sequential resampling is unsuitable for testing statistical hypotheses. There remains the big question in the daily practice of chemometrics and in QSAR and 3D QSAR designs of how to select randomly the number of groups and their sizes. However, sequential resampling may be useful for diagnostic statistics which prove the assumptions of the underlying theory of hypothesis testing.

• Manne, R., Shen, H., & Liang, Y. (1999).
  Subwindow factor analysis. Chemometrics and Intelligent Laboratory Systems, 45, 171-176.
  The method of subwindow factor analysis (SFA) is introduced as a solution to the problem of directly extracting component spectra from overlapping structures obtained from hyphenated chromatography without first resolving concentration profiles. This is of advantage when a complete resolution cannot be obtained or is of less interest in the analytical situation. The method is based upon comparisons of chromatographic regions (subwindows) which have only one eluting component in common. The paper presents the theory and an application to a structure with 4 overlapping components from a data set from a mixture of polyaromatic hydrocarbons recorded by high-performance liquid chromatography diode array detection (HPLC-DAD).

• Mansurov, H.H., Muratov, F.H., & Timerkaev, V.S. (1986).
  Computational methods of the data-cube analysis in the chronic hepatic lesion research. Paper presented at COMPSTAT 86, Rome, 1-5 September.
  The paper discusses how information obtained from patients with chronic hepatic lesions at several points in time can be shaped into a three-way array so that it can be analyzed with three-way methods.

• Marasinghe, M.G., & Boik, R.J. (1993).
  A three-degree of freedom test of additivity in three-way classifications. Computational Statistics & Data Analysis, 16, 47-61.
  This article proposes a new interaction model for nonreplicated three-way classifications. A simulation study is used to show that a three-degree of freedom score test based on the new model compares favorably with existing one-degree of freedom score and likelihood ratio tests of additivity. The tests are illustrated through an analysis of a data set where it is shown how the new model may reveal a specific structure of three-factor interaction. This structure may be exploited to suggest possible explanations for the nonadditivity.

• Marchand, B. (1984).
  Urban growth models revisited: Cities as self-organizing systems. Environment and Planning A, 16, 949-964.
  Contains a summary of a three-mode analysis á la Tucker (1966) presented in full in Marchand (1986) "The Emergence of Los Angeles: Population and Housing in the City of Dreams, 1940-1970". Data from four censuses were arranged in a parallelepiped that has been analyzed through three-way factor analysis. The method, taken from Tucker (1966), is somewhat complex, but essentially leads to a description of change over time.

• Marcos, A., Foulkes, M., & Hill, S. J. (2001).
  Application of a multi-way method to study long-term stability in ICP-AES. Journal of Analytical Atomic Spectrometry,16, 105-114.
  Although major advantages have been made in developing robust, easy-to-use ICP-AES instruments offering sub mug g(-1) detection limits and relative interference free operation, long-term drift of the analytical signal continuous to be problematic and necessitates regular re-calibration. The work presented here focuses on the effect of two instrumental parameters, i.e. the rf power and the nebuliser gas flow rate, on the robustness of the signals. The effects on the long-term stability when varying these two factors was systematically studied using an experimental design protocol. A "drift diagnosis" on thirty emission lines was performed at 12 different sets of operating conditions by repeated determination of a multi-element solution over several hours. The results were studied using standard parameters, i.e., Mg ratio, sensitivity, drift error, drift patterns and multi-way analysis. Parallel factor analysis (PARAFAC) was employed to analyse the 3-way data array generated: "emission lines x replicates x operating conditions". The physical interpretation of the new PARAFAC-factors is shown to enable a better understanding of the drift phenomenon by mathematically characterising the causes of long-term instability. Finally, the robustness of the technique using different operating conditions is evaluated and the appropriate use of internal standards to correct for drift is discussed.

• Marengo, E., Leardi, R., Robotti, E., Righetti, P. G., Antonucci, F. & Cecconi, D. (2003).
  Application of three-way principal component analysis to the evaluation of two-dimensional maps in proteomics Journal of Proteome Research, 2, 351-360.
  Three-way PCA has been applied to proteomic pattern images to identify the classes of samples present in the dataset. The developed method has been applied to two different datasets: a rat sera dataset, constituted by five samples of healthy Wistar rat sera and five samples of nicotine-treated Wistar rat sera; a human lymph-node dataset constituted by four healthy lymph-nodes and four lymph-nodes affected by a non-Hodgkin's lymphoma. The method proved to be successful in the identification of the classes of samples present in both of the groups of 2D-PAGE images, and it allowed us to identify the regions of the two-dimensional maps responsible for the differences occurring between the classes for both rat sera and human lymph- nodes datasets.

• Marsili, N. R., Lista, A., Band, B. S. F., Goicoechea, H. C., & Olivieri, A. C. (2004).
  New method for the determination of benzoic and sorbic acids in commercial orange juices based on second-order spectrophotometric data generated by a pH gradient flow injection technique. Journal of Agricultural and Food Chemistry, 52, 2479-2484.
  Two widely employed antimicrobials, benzoic and sorbic acids, were simultaneously determined in commercial orange juices employing a combination of a flow injection system with pH gradient generation, diode array spectro photometric detection, and chemometric processing of the recorded second-order data. Parallel factor analysis and multivariate curve resolution-alternating least-squares were used for obtaining the spectral profiles of sample components and concentration profiles as a function of pH, including provisions for managing rank-deficient data sets. An appropriately designed calibration with a nine-sample set of binary mixtures of standards, coupled to the use of the second-order advantage offered by the applied chemometric techniques, allowed quantitation of the analytes in synthetic test samples and also in commercial orange juices, even in the presence of unmodeled interferents (with relative prediction errors of 8.7% for benzoic acid and 2.5% for sorbic acid). No prior separation or sample pretreatment steps were required. The comparison of results concerning commercial samples with a laborious reference technique yielded satisfactory statistical indicators (recoveries were 99.0% for benzoic acid and 101.4% for sorbic acid).

• Martens, H. (1980).
  On the calibration of a multivariate instrument for quantitative estimation of individual components in a mixture. In A. Höskuldsson, K. Conradsen, B.S. Jensen, & K. Esbensen (Eds.). Symposium I Anvendt Statistik [Symposium on Applied Statistics] (pp. 393-414). Lyngby, Denmark: Technical University of Denmark.
  Within the context of indirect calibration based on extensions of Beer's law, Martens discusses the potential use of three-way methods for calibration with hyphenated instruments. In particular, the attractiveness of the PARAFAC model (discussed within the context of Sands & Young's (1980, 1983) ALSCOMP3) is indicated for spectometric data, emphasizing the uniqueness of the model.

• Martínez-Montes, E., Valdés-Sosa, P. A., Miwakeichi, F., Goldman, R. I., & Cohen, M. S. (2004).
  Concurrent EEG/fMRI analysis by multiway Partial Least Squares. NeuroImage, 22, 1023-1034.
  Data may now be recorded concurrently from EEG and functional MRI, using the Simultaneous Imaging for Tomographic Electrophys-iology (SITE) method. As yet, there is no established means to integrate the analysis of the combined data set. Recognizing that the hemodynamically convolved time-varying EEG spectrum, S, is intrinsically multidimensional in space, frequency, and time motivated us to use multiway Partial Least-Squares (N-PLS) analysis to decompose EEG (independent variable) and fMRI (dependent variable) data uniquely as a sum of ‘‘atoms’’. Each EEG atom is the outer product of spatial, spectral, and temporal signatures and each fMRI atom the product of spatial and temporal signatures. The decomposition was constrained to maximize the covariance between corresponding temporal signatures of the EEG and fMRI. On all data sets, three components whose spectral peaks were in the theta, alpha, and gamma bands appeared; only the alpha atom had a significant temporal correlation with the fMRI signal. The spatial distribution of the alpha-band atom included parieto-occipital cortex, thalamus, and insula, and corresponded closely to that reported by Goldman et al. [NeuroReport 13(18) (2002) 2487] using a more conventional analysis. The source reconstruction from EEG spatial signature showed only the parieto-occipital sources. We interpret these results to indicate that some electrical sources may be intrinsically invisible to scalp EEG, yet may be revealed through conjoint analysis of EEG and fMRI data. These results may also expose brain regions that participate in the control of brain rhythms but may not themselves be generators. As of yet, no single neuroimaging method offers the optimal combination of spatial and temporal resolution; fusing fMRI and EEG meaningfully extends the spatio-temporal resolution and sensitivity of each method.

• Martins, J.A., Sena, M.M., Poppi, R.J. & Pessine, F.B.T. (1999).
  Fluorescence Piroxicam study in the presence of cyclodextrins by using the PARAFAC method. Applied Spectroscopy, 53, 510-522.
  The nonsteroidal anti-inflammatory drug Piroxicam in solution presents several equilibria that depend on its conformational structure and the medium. It was studied by total fluorescence by using parallel factor analysis (PARAFAC) to resolve its spectra. It was possible to identify a correlation between some spectra and the isomers and to reach some understanding of the spectroscopic behaviour of the drug in cyclodextrin-containing hydrophilic solvents.

• Mastandrea, S., Zani, A., Giuliani, M. V., & Bove, G. (1993).
  Meaning of industrial-design objects - from designers to users. Environment and Planning B-Planning & Design, 20, 307-319.
  Structural features of everyday objects of industrial design and expressive qualities possibly communicated by the same objects are investigated. The objective may be defined in terms of the following points: (1) communication between designers and users; (2) differences in appraisal between groups of experts and nonexperts; (3) systematic relationships between structural characteristics and expressive qualities of the objects considered. Three groups of subjects were interviewed: four designers responsible for the design of six objects, twenty advanced-level design students (experts), and twenty nonexpert students. All subjects had to fill in a questionnaire based on an open interview with the designers. The questionnaire was divided into two parts: structural characteristics and expressive characteristics. A multifactorial analysis and t-test were performed on the data. The results suggest that (1) communication between designers and users exists in a large number of item appraisals and is not the result of the ambiguity of the physical properties of the objects; (2) specific training in design has a direct influence upon object appraisal, indicating a certain differentiation between the groups of experts and nonexperts; (3) there are no systematic correlations between structural and expressive characteristics except in one very specific case: between the structural characteristic of shape and consistency of material, on the one hand, and the expressive qualities of dynamism, on the other.

• Mayekawa, S. (1987).
  Maximum likelihood solution to the PARAFAC model. Behaviormetrika, 21, 45-63.
  The traditional factor analytic view of the PARAFAC model and its extension to a four mode situation with the derivation of the maximum likelihood estimation procedure by the generalized EM algorithm was presented. The four mode model was applied to six ASVAB data matrices defined by three specialty (clerical, mechanical, and electrical), times two services, (Air Force and Marine Corps) and successfully recovered the usual four dimensional structure without any rotation. The specialty and service differences was expressed in terms of different weighting of the common factor structure. A model which allows us to compare the factor score means was also investigated.

  
     

• McAdams, S., Winsberg, S., Donnadieu, S., De Soete, G., & Krimphoff, J. (1995).
  Perceptual scaling of synthesized musical timbres: Common dimensions, specifities, and latent subject classes. Psychological Research, 58, 177-192.
  To study the perceptual structure of musical timbre and the effects of musical training, timbral dissimilarities of synthesized instrument sounds were rated by professional musicians, amateur musicians, and nonmusicians. The data were analyzed with an extended version of the multidimensional scaling algorithm CLASCAL (Winsberg & De Soete, 1993), which estimated the number of latent classes of subjects, the coordinates of each timbre on common Euclidean di- mensions, a specificity value of unique attributes for each timbre, and a separate weight for each latent class on each of the common dimensions and the set of specificities. Five latent classes were found for a three-dimensional spatial model with specificities. Weight patterns indicate that perceptual salience of dimensions and specificities varied across classes. The model with latent classes and specificities gave a better fit to the data and made the acoustic correlates of the common dimensions more interpretable.

  

• McArdle, J.J. (1984).
  On the madness in his method: R.B. Cattell's contributions to structural equation modeling. Multivariate Behavioral Research, 19, 245-267.
  This essay looks at the many methodological contributions of R.B. Cattell from the vantage point of contemporary issues in structural equation modeling. Cattell's factor analytical approach is compared with current modeling practices. A critical evaluation is offered which finds much of Cattell's work still innovative, still technically advanced, and still of great value to contemporary model builders.

• McArdle, J. J. & Cattell, R. B. (1994).
  Structural equation models of factorial invariance in parallel proportional profiles and oblique confactor problems. Multivariate Behavioral Research, 29, 63-113.
  Some problems of multiple group factor rotation based on Cattell's ''parallel proportional profiles'' and ''confactor rotation'' are described (see Cattell, 1944, 1966, 1972). Some relations between these classic ideas and contemporary practices in structural equation modeling (e.g., LISREL) are explored. We show how the Confactor approach: (a) is related to Meredith's (1964a) selection model, (b) can be a parsimonious model for multiple group factor analyses, and (c) how this model can be fitted using standard structural equation modeling techniques- We discuss several alternative structural modeling solutions, including (d) selection of a good reference variable solution, (e) rotation of the invariant orthogonal structure by standard rotation routines, and (f) higher-order, latent paths, and latent means structural model restrictions. Mathematical and statistical properties of these models are examined using Meredith's (1964b) four group problem fitted by Joreskog and Sorbom's (1979, 1985) LISREL algorithm. The benefits and limitations of this structural modeling approach to oblique Confactor resolution are examined and opportunities for future research are discussed.

• McCloskey, J. & Jackson, P.R. (1979).
  THREE-MODE: A FORTRAN IV program for three-mode factor analysis. Behavior Research Methods & Instrumentation, 11, 75-76.
  A computer program for T3 (Tucker, 1966 - Method I) is described without technical details.

• McDonald, R.P. (1969).
  A generalized common factor analysis based on residual covariance matrices of prescribed structure. The British Journal of Mathematical and Statistical Psychology, 22, 149-163.
  An algorithm is described, the purpose of which is to express a correlation matrix as the sum of two matrices, of which one is of rank less than its order, and the other, the residual matrix, is of prescribed structure. Applications are described to approximate simplex and circumplex matrices, to the factoring of groups of variables, and to a form of multi-mode factor analysis.

  
     

• Medina, R., Losada, M.A., Losada, I.J., & Vidal, C. (1994).
  Temporal and spatial relationship between sediment grain-size and beach profile. Marine Geology, 118, 195-206.**
  Sediment samples and beach profile evolution data collected along one profile line at ''El Puntal'' spit, Santander, Spain, are used to analyze the spatial and temporal structure of the grain size distribution variability and its relationship with the beach profile changes. Standard principal component analysis (PCA) and three-way PCA is applied to determine the temporal and spatial scales of variability of the data. Results indicate that the sediment grain size distribution varies markedly along the beach profile both spatially and temporally. These variations are shown to be strongly related to morphological changes in the beach profile. The spatial eigenvectors determined from the profile data and those from the sediment data exhibit similar patterns with their maxima and minima located at the same position. Since eigenvectors may be regarded as representative of uncorrelated modes of variability it is concluded that the spatial variability of both sediment and profile data are strongly related. In particular, it is shown that the location of the highest variability of grain size corresponds to that of the beach profile. Also, different grain sizes are shown to exhibit a distinct degree of variability which leads to the conclusion that each sediment size responds to the same hydrodynamics differently. The temporal eigenvectors determined from the profile and the sediment data shown a seasonal dependency. However, their maxima and minima are not located at the same position. It is shown that this temporal shift is due to the different response of each sediment size to the hydrodynamics, and in particular, that the recovery of the profile starts with fine material from the bar. It is inferred that models for beach profile evolution which do not take into account the sorting processes involved in the sediment transport cannot be fully succesful. A "master" grain size sample, constructed by adding all the grain samples taken over the profile, is used to further examine the cross-shore redistribution of the sediment. The following working hypothesis is suggested: "For a beach profile within a physiographic unit the master grain size does not depend on time".

• Mendieta, J., Díaz-Cruz, M. S., Esteban, M., & Tauler, R. (1998).
  Multivariate Curve Resolution: A Possible Tool in the Detection of Inermediate Structures in Protein Folding. Biophysical Journal, 74, 2876-28888.
  Different multivariate data analysis techniques based on factor analysis and multivariate curve resolution are shown for the study of biochemical evolutionary processes like conformational changes and protein folding. Several simulated CD spectral data sets describing different hypothetical protein folding pathways are analyzed and discussed in relation to the feasibility of factor analysis techniques to detect and resolve the number of components needed to explain the evolution of the CD spectra corresponding to the process (i.e., to detect the presence of intermediate forms). When more than two components (the native and unordered forms) are needed to explain the evolution of the spectra, an iterative multivariate curve resolution procedure based on an alternating least squares algorithm is proposed to estimate the CD spectrum corresponding to the intermediate form.

• Meijs, B.W.G.Ph. (1980).
  Huis van bewaring en subkultuur: een empirische studie bij jeugdige gedetineerden naar het effekt van '102 dagen' preventieve hechtenis op attitudes en andere indikatoren van subkultuur. Unpublished master thesis, University of Leiden, The Netherlands.
  T3 (as implemented by Kroonenberg & De Leeuw, 1980) was one of the analyses of semantic differential data from 37 juvenile delinquents judging their surroundings, i.e. remand prison, home, law enforcers, and the judiciary (26 aspects, 10 scales). The data were collected at two time point (directly after intake and one month later), and each time point was analysed separately. Interesting shifts of opinions about aspects could be observed.

• Meng, X., Morris, A. J. & Martin, E. B. (2003).
  On-line monitoring of batch processes using a PARAFAC representation Journal of Chemometrics, 17, 65-81.
  For assured through-batch process performance monitoring, a number of established bilinear and trilinear modelling techniques require data to be available for the entire duration of the batch to realize the on-line application of the nominal model. Various strategies have been proposed for the in-filling of those yet unknown values. A methodology is presented where the unknown observations are calculated as a weighted combination of the, scores up to the current time. point in the new-batch and those previously computed from a reference data set. This approach is investigated for the trilinear technique of parallel factor analysis (PARAFAC). Modified confidence limits are then proposed for the bivariate scores plot for on-line monitoring with a PARAFAC model. The identification of those variables indicative of causing changes in process operation has been accomplished through the application of contribution plots. Based on such plots, a methodology, with associated confidence limits, is proposed for the location of those variables whose behaviour differs from that encapsulated within the reference data set. The approach is demonstrated and compared with existing techniques on a benchmark simulation of a semi-batch emulsion polymerization that has-been used in similar studies.

• Meulders, M., De Boeck, P., Kuppens, P., & van Mechelen, I. (2002).
  Constrained latent class analysis of three-way three-mode data. Journal of Classification, 19, 277-302.
  

• Meulman, J.J. (1989).
  Distance analysis in reduced canonical spaces. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp.233-244).
  Amsterdam: Elsevier.
  A particular form of generalized canonical analysis compares p- dimensional linear combinations of M sets of variables with a p-dimensional unknown matrix X. This technique can be shown to have certain distance properties in terms of the objects in the analysis. From this starting point various modifications of the technique can be introduced that optimize these distance properties. In this paper a quite general approach is presented that has a previous generalization (Meulman, 1986) as a special case.

  Meulman, J.J., & Verboon, P. (1993).
  Points of view analysis revisited: fitting multidimensional structures to optimal distance components with cluster restrictions on the variables. Psychometrika, 58, 7-35.
  A procedure is proposed that can be viewed as a streamlined, integrated version of the Tucker and Messick (1963) Point-of-View analysis, which consisted of a number of separate steps. At the same time, our procedure can be regarded as a particularly constrained weighted Euclidean model. While fitting the model, two types of nonlinear data transformations are feasible, either for given dissimilarities, where the two types of transformation can be mixed in the same analysis; a quadratic assignment framework is used to evaluate the results.

• Meuwese, W. (1970).
  Een vergelijking van twee methoden van beoordeling van verbale stimuli. Nederlands Tijdschrift voor de Psychologie, 25, 594-603.
  A selection of 15 university subjects were judged by 44 staff and students of a technical university on 21 seven-point rating scales. T3 was performed, and all modes were interpreted but not the core matrix. The stimulus space was compared with a MDS-solution obtained from paired comparison data.

  

• Meyer, J.P. (1980).
  Causal attribution for success and failure: A multivariate investigation of dimensionality, formation, and consequences.Journal of Personality and Social Psychology, 38, 704-718.
  The dimensions underlying causal attributions for success and failure, the influence of various informational cues on attributions along the dimensions, and the consequences of such attributions were investigated using PARAFAC (Harshman, 1976). Hypothetical cases describing high school students' performance on a university entrance exam and in high school, the performance of others on the exam, and the importance of the exam were presented to 193 male undergraduates in a within-Ss design.

• Meyer, P., Wilhelms, R., & Strube, H. W. (1989).
  A quasiarticulatory speech synthesizer for German language running in real-time. Journal of the Acoustical Society of America, 86, 523-539.
  A system for simple quasiarticulatory speech synthesis is described. It is based on an articulatory model, which is controlled by seven parameters. The synthesizer employs a stylized vocal tract model that is realized using wave digital filter techniques. It includes a self-oscillating glottis model, voiceless excitation in the tract, damping, a nasal tract, and a radiation load that allows the simulation of lip protrusion. The synthesizer runs in real time on a signal processor. The described model was fit to natural speech using a Kalman filter algorithm. Simple rules for the synthesis of unrestricted German speech are described.

• Meyners, M. (2001). Statistische Eigenschaften der STATIS-Methode -Propriétés statistiques de la méthode STATIS.[Statistical properties of the STATIS method]. Göttingen: Cuvilier.
  Various statistical properties of the STATIS-method were investigate. Main issues are the difference with a true underlying consensus solution and the asymptotic properties whenever the number of products and / or assessors converges to infinity. It was shown that STATIS overestimates dimensionality of the consensus. Therefore, a correction to STATIS was proposed in order to correct for this as much as possible. By means of simulations, this corrected version of STATIS was compared with the original version as well as with two additional variants, one of which uses asymptotically optimal weights for the assessors, the other one does not use weights. It was found that the corrected versions outperformed all others and, in an additional study, there were shown to be comparable to Generalized Procrustes Analysis (GPA). A disavantage of GPA is that it requires an iterative algorithm, while the corrected STATIS does not. Graphical comparisons supported the theoretical and empirical results, be it that, with respect to the interpretation of the results, all methods under consideration seemed to be comparable for the cases considered.

• Meyners, M. (2000).
  Comparing generalized procrustes analysis and STATIS. Food Quality and Preference 11 77-83.
  We consider a model for sensory profiling data including translation, rotation and scaling. We compare two methods to calculate an overall consensus from several data matrices: Generalized Procrustes Analysis (GPA) and STATIS (Structuration des Tableaux à Trois Indices de la Statistique). These methods are briefly illustrated and explained under our model. A series of simulations to compare their performance has been carried out. We found significant differences in performance depending on the variance of random errors and on the dimensionality of the true underlying consensus. Therefore, we investigated on the dimensionality of the calculated group averages. We found both methods gave too many dimensions compared to the true consensus. This finding is supported by some theoretical considerations. Finally we propose a combined approach which takes advantage of both methods and which gave better results in the simulations.

  
         

• Michailidis, G., & De Leeuw, J. (2000).
  Multilevel homogeneity analysis with differential weighting. Computational Statistics and Data Analysis, 32, 411-442.
  A model is introduced for multilevel homogeneity analysis with fixed group category quantification across groups with differential weights for each group. The authors show this model to be a constrained form of the standard PARAFAC model.

• Miettinen, T., Hurse, T. J., Connor, M. A., Reinikainen, S. P., & Minkkinen, P. (2004).
  Multivariate monitoring of a biological wastewater treatment process: a case study at Melbourne Water's Western Treatment Plant. Chemometrics and Intelligent Laboratory Systems, 73, 131-138.
  Biological wastewater treatment is a complex, multivariate process, in which a number of physical and biological processes occur simultaneously. In this study, principal component analysis (PCA) and parallel factor analysis (PARAFAC) were used to profile and characterise Lagoon 115E, a multistage biological lagoon treatment system at Melbourne Water's Western Treatment Plant (WTP) in Melbourne, Australia. In this study, the objective was to increase our understanding of the multivariate processes taking place in the lagoon. The data used in the study span a 7-year period during which samples were collected as often as weekly from the ponds of Lagoon 115E and subjected to analysis. The resulting database, involving 19 chemical and physical variables, was studied using the multivariate data analysis methods PCA and PARAFAC. With these methods, alterations in the state of the wastewater due to intrinsic and extrinsic factors could be discerned. The methods were effective in illustrating and visually representing the complex purification stages and cyclic changes occurring along the lagoon system. The two methods proved complementary, with each having its own beneficial features.

• Miller, K., & Gelman, R. (1983).
  The child's representation of number: A multidimensional scaling analysis. Child Development, 54, 1470-1479.
  In order to describe developments in children's conceptions of number, judgments of similarities between numbers were solicited from children in grades kindergarten, 3, and 6 as well as from adults. Analysis of the resulting data by clustering and nonmetric multidimensional scaling techniques suggested that children become sensitive to an expanding set of numerical relations during this period, although even kindergartners appear to understand the importance of magnitude as a basis for judging similarity between numbers. This study is an application of INDCLUS.

• Millican, D.W., & McGown, L.B. (1990).
  Fluorescence lifetime resolution of spectra in the frequency domain using multiway analysis. Analytical Chemistry, 62, 2242-2247.
  Spectral resolution of two-component systems by multiway analysis of excitation-emission-frequency arrays is compared with single-matrix analysis of steady-state excitation-emission matrices for both real and computer-simulated systems. Fluorescence lifetime selectivity can improve resolution, especially for the minor component in mixtures in which the two components have unequal contributions to the total intensity. However, in some cases the lifetime difference between two fluorophores is insufficient to aid resolution.

• Mills, D.H. & Tucker, L.R. (1966).
  A three-mode factor analysis of clinical judgment of schizophrenicity. Journal of Clinical Psychology, 22, 136-139.
  Schizophrenicity judgments of 23 persons (patients and normals) on 20 items of the WAIS' vocabulary and comprehension by 5 judges was subjected to T3. A nice, clearly interpretable solution appeared.

• Millsap, R.E., & Meredith, W. (1991).
  Some mathematical relationships between three-mode component analysis and stationary component analysis. Multivariate Behavioral Research, 26, 413-419.
  One application of three-mode component analysis is in the analysis of longitudinal multivariate data. Millsap and Meredith (1988) have developed a component analysis method for this application that uses stationary compositing weights. This article presents theorems giving constraints which must be satisfied for equivalency between the component representations provided by three- mode component analysis and by methods employing stationary compositing weights. In general, the two approaches will provide mathematically distinct representations.

  

• Mirkin, B.G. (1987).
  Additive clustering and qualitative factor analysis methods for similarity matrices. Journal of Classification, 4, 7-31.
  This paper reviews methods of qualitative factor analysis (QFA) developed by the author and his collaborators over the last decade and discusses the use of QFA methods for the additive clustering problem. The QFA method includes, first, finding a square Boolean matrix in a fixed set of Boolean matrices with "simple structure" to approximate a given similarity matrix, and, second, repeating this process again and again using residual similarity matrices. Convergence properties for three versions of the method are presented, "cluster" interpretations for results obtained from the algorithms provided, and formulas for the evaluation of "factor shares" of the initial similarities variance given.

• Mitchell, B.C., & Burdick, D.S. (1993).
  An empirical comparison of resolution methods for three-way arrays. Chemometrics and Intelligent Laboratory Systems, 20, 149- 161.**
  In chemometrics applications it is common to resolve a trilinear array by solving a generalized eigenvalue problem, rather than by employing the iterative parafac algorithm commonly used by psychometricians. Although an eigenanalysis-based procedure works perfectly in the absence of noise, it is not guaranteed to yield least squares resolutions when noise is present. The Parafac algorithm on the other hand is guaranteed to reduce the residual sum of squares at each iteration. In this paper we propose synthesizing the two methods by using the resolution generated by eigenanalysis as starting values for the iterative Parafac algorithm. We find for simulated four-component data at moderate noise levels that following an eigenanalysis resolution with Parafac frequently leads to significant improvement in the quality of the resolution.

• Mitchell, B.C., & Burdick, D.S. (1994).
  Slowly converging Parafac sequences: Swamps and two-factor degeneracies. Journal of Chemometrics, 8, 155-168.
  The alternating least squares Parafac algorithm is a useful tool for resolving trilinear three-way data arrays. Occasionally, however, it becomes bogged down for many iterations in the vicinity of a poor quality resolution before moving on to a much superior optimum fit. We investigate this behavior in a simulation study and suggest ways of overcoming the obstacles it presents.

• Miyano, H., & Inukai, Y. (1982).
  Sequential estimation in multidimensional scaling. Psychometrika, 47, 321-336.
  The concept of sequential estimation is introduced in multidimensional scaling (MDS). The sequential estimation method developed in this paper refers to continually updating estimates of a configuration as new observations are added. This method has a number of advantages, such as a locally optimal design of the experiment can be easily constructed, and dynamic experimentation is made possible. Using artificial data, the performance of our sequential method is illustrated.

• Miwakeichi, F., Martinez-Montes, E., Valdes-Sosa, P. A., Nishiyama, N., Mizuhara, H., & Yamaguchia, Y.(2004).
  Decomposing EEG data into space-time-frequency components using Parallel Factor Analysis. Neuroimage, 22, 1035-1045.
  Finding the means to efficiently summarize electroencephalographic data has been a long-standing problem in electrophysiology. A popular approach is identification of component modes on the basis of the time-varying spectrum of multichannel EEG recordings-in other words, a space/frequency/time atomic decomposition of the time-varying EEG spectrum. Previous work has been limited to only two of these dimensions. Principal Component Analysis (PCA) and Independent Component Analysis (ICA) have been used to create space/time decompositions; suffering an inherent lack of uniqueness that is overcome only by imposing constraints of orthogonality or independence of atoms. Conventional frequency/time decompositions ignore the spatial aspects of the EEG. Framing of the data being as a three-way array indexed by channel, frequency, and time allows the application of a unique decomposition that is known as Parallel Factor Analysis (PARAFAC). Each atom is the tri-linear decomposition into a spatial, spectral, and temporal signature. We applied this decomposition to the EEG recordings of five subjects during the resting state and during mental arithmetic. Common to all subjects were two atoms with spectra signatures whose peaks were in the theta and alpha range. These signatures were modulated by physiological state, increasing during the resting stage for alpha and during mental arithmetic for theta. Furthermore, we describe a new method (Source Spectra Imaging or SSI) to estimate the location of electric current sources from the EEG spectrum. The topography of the theta atom is frontal and the maximum of the corresponding SSI solution is in the anterior frontal cortex. The topography of the alpha atom is occipital with maximum of the SSI solution in the visual cortex. We show that the proposed decomposition can be used to search for activity with a given spectral and topographic profile in new recordings, and that the method may be useful for artifact recognition and removal.

  

• MiyanoH. (1996).
  Three-way data analysis based on Tucker2 model: A method utilizing a priori information on some model parameters. Japanese Psychological Review, 39,429-438 (in Japanese with English abstract).
  It is well-known that each of the component matrices in the Tucker2 model is only determined upto a nonsingular transformation. This indeterminacy often makes it difficult to derive substantive information present in the data. A new algorithm is proposed for the Tucker2 model which utilises a priori information on the model parameters. The efficiency of the proposed algorith is confirmed by analysing both an artificial data set and real data.

• Mizère, D. (1981).
  Analyse d'un cube de données: décomposition tensorielle et liens entre procédures de comparaison de tableaux rectangulaires de données. [Analysis of a data cube: Tensorial decomposition and links with procedures to compare rectangular data matrices]. Unpublished doctoral thesis, The Scientific and Medical University of Grenoble, France.
  I. Algebraic background.
  Euclidian spaces: duality diagrams and metrics.
  II. Analysis with second order tensors.
  Treatment of three-way data with second order tensors.
  III. Analysis with third order tensors.
  Mathematical and statistical developments; Tensorial decomposition and principal component analysis.
  IV. Applications.
  Matrices as a specification of three-way arrays; Procrustes rotation and the search for a intrametric; Comparison with other procedures.

• Mizère, D. (1984).
  L'Aspect choix de métrique dans l'analyse d'un cube de contingence. Afrika Matematika, 6, 49-68.
  In this paper research into three-dimensional matrices is extended to specific contingency tables.

  
             

• Moberg, L., Robertsson, G., & Karlberg, B. (2001).
  Spectrofluorimetric determination of chlorophylls and pheopigments using parallel factor analysis. Talanta, 54, 161-170.
  In this study, parallel factor analysis (PARAFAC) was applied to fluorescence excitation emission matrices (EEM) of chlorophylls and pheopigments dissolved in acetone:water (9:1). The excitation wavelength range was from 350 to 500 nm and the emission was recorded from 600 to 730 nm. Nine standards, comprising mixtures of six analytes, were decomposed into a six-component PARAFAC model. Each component resembled the corresponding EEM of the pure analyte, demonstrating the uniqueness properties of PARAFAC. The score matrix obtained from the model was used for calibration and prediction of an independent set of standards and for eleven samples collected in the Baltic proper. The results obtained by the proposed method were compared to classical least squares (CLS) and to predictions by reference methods (HPLC and visible spectroscopy). For the independent set of standards the proposed method and CLS performed equal well in terms of predictive ability. But fur the samples the proposed method yielded results that were in good agreement to the reference methods, whereas CLS failed. Also the so-called "second-order advantage" was examined, showing that not all constituents must be included in the calibration set. The concentration range was for chlorophyll a varied between 10 and 75 mug l(-1) and similar for the other analytes.

• Moller, J. K. S., Parolari, G., Gabba, L., Christensen, J., & Skibsted, L. H. (2003).
  Monitoring chemical changes of dry-cured parma ham during processing by surface autofluorescence spectroscopy. Journal of Agricultural and Food Chemistry, 51, 1224-1230.
  Parma hams at various processing stages were investigated by surface autofluorescence spectroscopy. Fluorescence "landscapes" of raw meat and salted (3 months), matured (11 and 12 months), and aged (15 and 18 months) Parma hams were obtained, and a three-dimensional data array (sample x emission x excitation) was used to develop a PARAFAC model including five components, which all exhibited characteristics of pure fluorophores regarding both excitation and emission spectra. The relative amount of each component related strongly to the processing stage, and sample age showed good correlation to fluorescence data (R = 0.98), with a relative error of prediction of approximately 1 month. Fluorescence measurements from samples of either semimembranosus or biceps femoris were used to predict chemical or sensory reference data, yielding good correlation for biceps femoris data, thereby enabling moisture content, sensory and instrumental color, and proteolysis value to be fairly well predicted. Overall, surface autofluorescence of Parma hams proved to hold relevant information, relating to major chemical/physical changes during processing. It is concluded that fluorescence spectroscopy has potential as an innovative method of quality control in dry-cured ham.

• Möcks, J. (1988a).
  Decomposing event-related potentials: A new topographic components model. Biological Psychology, 26, 199-215.
  "Component" notions inherently used with measurement approaches, raw peak determination, PCA and generator approaches are discussed. By combining aspects of them all, a new model of ERP decomposition is established; quite profitable and surprising mathematical properties are illutrated and discussed.

  

• Möcks, J. (1988b).
  Topographic components model for event-related potentials and some biophysical considerations. IEEE Transactions on Biomedical Engineering, 35, 482-484.
  Models for decomposing averaged event-related potentials in component functions are discussed. Biophysical considerations motivate a new sample model, which is shown to lead to unique identifiable components, thereby overcoming a major drawback of the customary approach by principal components analysis.

• Montanelli, D.S. (1972).
  Multiple-cue learning in children. Developmental Psychology, 7, 302-312.
  Response patterns of 144 children in a multiple-cue learning experiment with 8 trial blocks and 3 cue weightings were ana- lysed by T3 (or actually T2, as no components were determined for the cue weightings). Very little numerical information was presented and only a cursory interpretation given.

• Montano,D. & Adamopoulos, J. (1984).
  The perception of crowding in interpersonal situations: Affective and behavioral responses.Environment and Behavior, 16, 643-66
  Likelihood judgments of responses of 280 undergraduates to crowded situations were subjected to three-mode factor analysis. The dimensionality of the components are discussed, and behavior patterns in different situational contexts are identified.

  

• Mooijaart, A. (1992).
  Three-factor interaction models by log-trilinear terms in three-way contingency tables. Statistica Applicata, 4, 669-677.
  Association models are usually defined for two-way tables. In this paper these models will be generalized to three-way tables, where the three-factor interaction parameters are constrained to be equal to one or more log-trililnear terms. The generalization is analogous to the PARAFAC/CANDECOMP model for continuous data. Parameters will be estimated by the maximum likelihood method. An example with real data will illustrate the usefulness of our model.

• Moonen, J. (1978).
  Computergestuurd Onderwijs. Een onderzoek naar de mogelijkheden tot geïntegreerd gebruik van een computergestuurd systeem in een statistiekkurrikulum (doctoral thesis). Leiden, The Netherlands: author.
  As part of a large scale investigation into the possibilities of integrating computer aided instruction into a statistics course, the data of 121 subjects rating 8 organisational approaches to studying statistics on 8 aspects related to a statistics course were analysed using TUCKALS2 (Kroonenberg & De Leeuw, 1977,1980, 1981c). The data were collected and analysed twice, once after one third of the course was over, and once at the end of the course. Rather straight forward interpretation; no interpretation was attempted of the subjects.

• Moreda-Pineiro, A., Marcos, A., Fisher, A., & Hill, S. J. (2001a).
  Chemometrics approaches for the study of systematic error in inductively coupled plasma atomic emission spectrometry and mass spectrometry. Journal of Analytical Atomic Spectrometry, 16, 350-359.
  Systematic errors observed when using inductively coupled plasma atomic emission spectrometry (ICP-AES) and mass spectrometry (ICP-MS) for the multi-element determination in acid digests of environmental samples (tea leaves) were evaluated. Two chemometric approaches, experimental design and principal component analysis, were used in order to establish the errors associated with each stage of the analytical method: sample digestion (effect of the "number of acid digestions") and measurement step (effect of the "number of replicates" and the "calibration"). The elements under study were Co, Cr, Cs, Cu, Ni, Pb, Rb and Ti by ICP-MS, and Ba, Ca, Fe, Mg, Mn, Sr and Zn by ICP-MS and ICP-AES. Flame atomic absorption spectrometry was used for comparative purposes. A Chinese tea certified reference material with certified concentration for most of the elements was employed as the sample matrix. Variance estimation was made from ANOVA outputs from a full factorial design (FFD) 4(1) x 4(1) x 2(1).

• Moreda-Pineiro, A., Marcos, A., Fisher, A., & Hill, S. J. (2001b).
  Parallel factor analysis for the study of systematic error in inductively coupled plasma atomic emission spectrometry and mass spectrometry. Journal of Analytical Atomic Spectrometry, 16, 360-369.
  This paper describes the application of a trilinear parallel factor analysis (PARAFAC) to study systematic error during the multi-element determination of a range of analytes in acid digests of solid samples (tea leaves) by ICP-AES and ICP-MS. The three variables studied were the "number of digestions", in order to assess the systematic error associated with the sample pre-treatment, and the "number of replicates" and "calibration", to provide information on the systematic error associated with the analytical determination itself. The elements under study were Co, Cr, Cu, Ni, Pb, Rb and Ti by ICP-MS, and Ba, Ca, Fe, Mg, Mn, Sr and Zn by both ICP-MS and ICP-AES. For some elements flame atomic absorption spectrometry was used for comparative purposes. A Chinese tea certified reference material containing many of the metals above was used in the study. The results obtained were compared to results from ANOVA. It was found that the systematic error, expressed as the sum of squares after PARAFAC, was quite different from the results obtained using ANOVA due to the very different way in which the models are built. The PARAFAC approach is shown to be straightforward to implement and robust.

• Morais, H., Ramos, C., Forgács, E., Cserháti, T., Oliviera, J., & Illés, T. (2001).
  Three-dimensional principal component analysis employed for the study of the beta-glucosidase prodution of Lentinus edodes strains. Chemometrics and Intelligent Laboratory Systems, 57, 57-64.
  The effect of cation type and concentration and fermentation time on the beta-glucosidase of four strains of Lentinus edodes was determined. Three-dimensional principal component analysis (3D-PCA) followed by nonlinear mapping technique (NLMAP) was employed for the assessment of similarities and dissimilarities between the enzymatic activities. The results proved that 3D_PCA followed by NLMAP is a valuable tool for the evaluation of three-dimensional data matrices in biology and microbiology.

• Morais, H., Ramos, C., Forgács, E., Jakab, A., Cserháti, T., Oliviera, J., Illés, T. & Illés, Z. (2003).
  Comparison of principal analysis and the Tucker3 model - A case study. QSAR & Combinatorial Science, 22, 449-455. A three-ways array data matrix consisting of the activity data of laccase enzyme has been evaluated by both principal component analyses (PCA) and Tucker3 model. Activity data have been determined in 28 culture media, at 6 sampling times and by four strains of Lentinus edodes. PCA has been carried out three times one of the factors being always the variables and the other two factors being the observations. The dimensionality of the matrices of loadings calculated by PCAs and those of component matrices of Tucker3 model has been reduced to two by the nonlinear mapping technique. It has been found that the dimensionality of component matrices for Tucker3 model can be predicted from the results of PCAs. Linear regression analyses indicated that the distribution of the original data points on the two-dimensional nonlinear maps considerably depends on the fact that the data have been calculated by PCA or by Tucker3 model.

• Morais, H., Ramos, C., Forgacs, E., Jakab, A., & Cserhati, T. (2004).
  Enzyme production of Pleurotus ostreatus evaluated by the three-way principal component analysis. Engineering in Life Sciences, 4, 165-170.
  The effect of the composition of culture media and fermentation time on the production of beta-glucosidase, xylanase, laccase, manganese-dependent and independent peroxidases by the edible fungus Pleurotus ostreatus was determined. Culture medium separately containing potato, pepper, and tomato extracts and the enzyme activities were assessed to 31.5 days of fermentation. Three-dimensional principal component analysis (3D-PCA) followed.by the plots of the first two elements of component matrices was employed for the elucidation of the similarities and dissimilarities between the parameters. It was established that the dimensionality of the original 3D matrix (9, 5, 4) could be reduced to 2, 1, 2 with 4.54 % loss of information. The plots demonstrated that the culture media displayed large differences in their capacity to promote enzyme production and that the presence of pepper and tomato extracts exerted the greatest influence on the enzyme activities. The effect of the fermentation time was manifested only after 1.4 d of fermentation and the highest differences were observed after 28 and 31.5 d of fermentation. Except laccase and manganese-dependent peroxidase, the enzyme activities also differed considerably dependent on the composition of the fermentation broth. 3D-PCA followed by the plots of component matrices is a valuable tool for the simultaneous assessment of three dimensional data matrices in biotechnological processes.

  
             

• Munck, L., Nørgaard, L., Engelsen, S.B., Bro, R., & Andersson, C.A. (1998). Chemometrics in food science - a demonstration of the feasibility of a highly exploratory, inductive evaluation strategy of fundamental scientific significance. Chemometrics and Intelligent Laboratory Systems, 44, 31-60.
  At the roots of science lies observation and data collection from the world as is and from which conclusions can be induced after classification. This is far from the present theory-driven, deductive, normative stage of science which depends heavily on modelling discrete functional factors in laboratory experiments and suppresses the aspect of interaction. In spite of its successes, science today has great difficulty in adapting to the changes which technology has created to cope with registering and evaluating real data from the world, such as in food production chains. This paper demonstrates that it is possible and profitable with the help of new technology to reintroduce an explorative, inductive strategy to investigate the chemistry of a complex food process as is with a minimum of a priori assumptions. The food process investigated is a sugar plant and the tools necessary in this strategy include a multivariate screening method (fluorescence spectroscopy), an arsenal of chemometric models (PCA, PLS, principal variables), including multiway models (Parafac, Tucker), and a computer. Not only can chemical criteria and process parameters throughout the process be validly predicted by the screening method, but process irregularities as well as chemical species can also be detected and validated by multiway chemometric techniques. Inspired by examples from the food area, the paper further discusses the nature of the exploration method in the selection of tools and data. The aim is to study complex processes as a whole in order to model interaction of the underlying latent functional factors which may later be defined more precisely by deductive methods. These methods in combination with an appropriate multivariate screening method allow for unique identification of objects - a significant prerequisite for a viable, exploratory, inductive data strategy which is needed as a fundamental complement to prevalent normative research in order to obtain a science on the interdisciplinary level.

• Murakami, T. (1979).
  Inshi henka no kijutsu to jun3so inshi bunseki [Description of factor change and quasi three-mode factor analysis]. Bulletin of the Faculty of Education, Nagoya University, 26, 1-16.
  A simplified three-mode factor analysis model is investigated as a procedure for factor analysis of data matrices obtained on several occasion and for assessment of factor change. The model, termed quasi three-mode factor analysis, approximates the raw data with a Tucker model. In the quasi three-mode model the two component matrices A and B are considered to be the criterion for description of factor change by core matrices. The proposed procedures do not yield the overall least squares solutions for Eq. (1), but make the quasi three- mode model interpretable as the summarized form of method I and method II. The core matrix G_k has a useful interpretation that it is the covariance matrix of factor scores in F_k with factor scores in A. Moreover, the core matrix can distinguish factor loadings change from factor scores change. As a matter of fact, G_k's are shown to exhibit the characteristic patterns corresponding to the four types of factor changes suggested by Baltes and Nesselroade (1973).

• Murakami, T. (1981).
  Saisho jijoho ni yoru jun3so inshi bunseki no algorithm [An algorithm of quasi three-mode factor analysis by the least squares methods]. Bulletin of the Faculty of Education, Nagoya University, 28, 39-59.
  Quasi three-mode factor analysis is a method for the factor analytic treatment of a 3-way data matrix. In this paper, a rationale of least squares algorithm for the model was formulated in the manner of the alternating least squares method and a practical computational procedure requiring only the multitrait- multimethod type correlation matrix is developed. This algorithm not only gave the least squares solution of quasi three-mode factor analysis in the strict sense, but also prompted the interpretation of the parameters. A slightly modified model in which the core matrix is transformed into the correlation matrix rather than the covariance matrix is discussed. The core matrix in this revised model can be regarded as a sort of higher order loading matrix, for its elements are the correlations of the factor scores for each occasion with the factor scores in common which might be, so to speak, the factor of factors. Kelly and Fiske's multitrait-multimethod matrix including five personality traits (Assertive, Cheerful, Serious, Unshakable poise and Broad interests) rated by three methods was reanalyzed in this model.

• Murakami, T. (1983a).
  3so data ni okeru inshi henka no kijutsu no tameno sho hoho [Methods for description of factor change in three-mode data (I): Factor analytical models]. Bulletin of the Faculty of Education, Nagoya University, 30, 145-176.
  This paper presents and compares several factor analytic models, some of which are new, dealing with three-mode data, especially longitudinal or multitrait- multimethod data. The main models are: General model; Longitudinal factor analysis model (Corballis, 1973); Multimethod factor analysis model (Jackson, 1975); Common factor loading model; Invariant factor loading model (Hakstian, 1973); Quasi three-mode factor analysis; Extended PARAFAC model; Common factor score model; Invariant model. Analysis of Kelly ∧ Fiske's multitrait- multimethod matrix is attached as a numerical example, and the type of the factor change described by each model is discussed.

• Murakami, T. (1983b).
  Quasi three-mode principal component analysis - A method for assessing the factor change. Behaviormetrika, 14, 27-48.
  Quasi three-mode factor analysis (= Tucker2 model) is described as a procedure for assessing factor change and it is shown to be applicable to longitudinal data and multitrait-multimethod matrices. The standardised data matrix for each occasion is decomposed as the product of three matrices, the factor score matrix, a slice of the extended core matrix, and the factor loading matrix. It is illustrated that the patterns to which core matrices conform can distinguish the change in factor loadings from the change in factor scores. A least squares algorithm for the model is formulated, and a way of scaling the parameters for easy interpretation and comparison with other factor analytic methods is discussed. The procedure is illustrated with a multitrait-multimethod matrix.

  

• Murakami, T. (1985).
  Tashugo data no tameno tansakuteki inshi bunseki [Exploratory factor analysis of multiset data]. Bulletin of the Faculty of Education, Nagoya University, 32, 75-93.
  A new exploratory method, multiset factor analysis, which factor analyzes data consisting of several sets of variables related to different domains or levels was proposed. An alternating least squares algorithm is formulated. This algorithm is a natural extensions of the ordinary principal factor method. Questionnaire data collected from housewives in Aichi prefecture was analyzed by this method. It consists of four sets, - buying behavior, attitude for buying, need for information and confidence in the various media. A factor which could not be found by separate factor analysis was derived in the first set and interesting relations between the factor and factors of other sets were revealed.

• Murakami, T. (1989).
  Theory of psychological measurement and the concept of family resemblances. Bulletin of the Faculty of Education, Nagoya University, 36, 149-156. (in Japanese, English abstract)
  The concept of unidimensionality is a central notion of classical test theory. As is well known, there is a dilemma between reliability and validity of a test because it is impossible to maximize both the coefficient alpha, an increasing function of variance of the test score, and the validity coefficient, a decreasing function of it simultaneously. But the dilemma is diminished if scores of all the items of the test are explained by a single factor in the basic model of factor analysis. Under the condition, it is demonstrated that both the correlation of an item with another item in the test and the correlation of the item with an external criterion are proportional to the factor loading of the item. In a way, the assumption of unidimensionality guarantees the legitimacy of ordinary item analysis technique.##

• Murakami, T. (1990).
  On the principal component analysis of multigroup, multiset, and multimode data. Kodokeiryugaku (Japanese Journal of Behaviormetrics), 18, 28-40 (in Japanese)
  This paper considers exploratory methods for investigating the factor invariance or factor changes in multigroup, multiset, and three-mode data, orthogonal rotations achieving the congruence of loading matrices, canonical analysis of component scores, and simultaneous principal component analytical methods with various optimization criteria and constraints. Desirable properties which should be shared by exploratory factor analytical methods are discussed based on simple schematic models of differences of factor loadings and factor scores as an extension of van de Geer's Maxbet method for multiset data. The method is equivalent to the slightly generalized Tuckals2 model (Kroonenberg and De Leeuw) for three-mode data, and can be modified to be applicable to multigroup data. It is demonstrated that these methods have the desirable properties as exploratory analysis.

• Murakami, T. (1991).
  Hierarchical principal component analysis of multiset-multigroup data. Bulletin of the Faculty of Education, Nagoya University, 38, 155-166. (in Japanese).
  Consider a data matrix Z, which is of the form: variables × subjects, and it can be partitioned into m sets of rows and g groups of columns. Z can then be seen as a super matrix. Let us call the data which can be arranged as Z multiset-multigroup one. This paper proposes a method for component analysis of multiset-multigroup data. The basic model is a natural extension of Kroonenberg & De Leeuw (1977)'s TUCKER 2 model for three-mode data. An alternating least squares algorithm, which is also a slight modification of TUCKALS 2 solving TUCKER 2 problem, is derived and adapted to handle the data with large N_s's. One of the most distinct features of the output of this method is that the first order loading matrices can be interpreted as correlation matrices between variables and first order components. This hierarchical component model is not only able to explain the data more parsimoniously than individual analysis of each element matrix but it is also more sensitive to the group differences of loadings than analysis of all groups as a whole. An application for the data with two sets - Peer and Self ratings, and two groups - males and females is demonstrated as an illustrative example.

• Murakami, T. (1996).
  Three-mode factor models and the concept of individual differences. Japanese Psychological Review, 39, 408-428 (in Japanese with English abstract).
  Two kinds of individual differences are found in the history of psychology, and distinguishing them is essential for the appropriate use of factor analytic three-mode models. Each type of individual difference correspondds to the specific way of preprocessing data, especially the mode across which is centred creates substantial differences in analytical results. There are at least two models: (1) individual differences models which embody differences in cognitive structures present in the individuals' factor structures, and which require three-mode data for their description; (2) condition differences models, in which individual differ in their location on dimensions, for instance, defined by test scores, and which can be described with two-mode data sets. However, three-mode models can be used to describe the variability and stability on the individual differences on several dimensions.

• Murakami, T. (1998).
  Tucker2 as a second-order principal component analysis. In C. Hayashi, N. Ohsumi, K. Yajima, Y. Tanaka, H.-H. Bock, & Y. Baba (Eds.), Data science, classification, and related methods (pp. 575-586). Tokyo: Springer.
  Statistical properties of the Tucker2 (T2) model, a simplified version of three-mode principal component analysis (PCA), are investigated aiming at applications to the study of factor invariance. The T2 model is derived as a restricted form of second-order PCA in the situation comparing component loadings and component scores across occasions. Several statistical interpretations of coefficients obtained from the least squares algorithm of T2 are proposed, and several aspects of T2 are shown to be natural extensions of characteristics of classical PCA. A scale free formulation of T2 and a new derivation of the algorithm for the large sample case are also shown. The relationship with a generalized canonical correlation model is suggested.

• Murakami, T. & Kroonenberg, P. M. (2003).
  Three-mode models and individual differences in semantic differential data. Multivariate Behavioral Research. 38 247-283.
  This article investigates how individual differences in semantic differential data can be modeled and assessed using three-mode models. Individual differences are important because their existence may affect the generality of conclusions based on such data. An overview is given how individual differences arise and how they can be handled in the analysis. The results of the investigation will be illustrated with semantic differential data on the characterization of Chopin's Preludes by a group of Japanese university students.

• Murakami, T., Ten Berge, J.M.F., & Kiers, H.A.L. (1998).
  A case of extreme simplicity of the core matrix in three-mode principal components analysis. Psychometrika, 63, 255-261.
  In three-mode Principle Components Analysis, the P × Q × R core matrix G can be transformed to simple structure before it is interpreted. It is well-known that, when P = QR, G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, when P = QR - 1, G can be transformed to have nearly all elements equal to values specified a priori. A closed- form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation of G are discussed.

• Murdoch, V.J., Snyder Jr, C.W., Law, H.G., Pamment, P.R., & Payne, P.V. (1985).
  Environmental evaluations of a "hard hat" residential area in rural Australia. Journal-of-Community-Psychology, 13, 360-374.
  244 construction workers and wives living in Nanango, a remote Austalian community, provided survey self-report ratings of (a) the availability of environmental features, (b) their importance to overall life quality, and (c) the extent to which they bring happiness and satisfaction. L. R. Tucker's (1966) 3-mode common factor analysis method was used to explicate the individual differences in environmental evaluations among these residents. Results suggest that Tucker's 3-mode factor analysis method was a useful way of partitioning individual differences' variance in environmental evaluations.

  

• Muroski, A.R., Booksh, K.S., & Myrick, M.L. (1996).
  Single-measurement excitation/emission matrix spectrofluorometer for determination of hydrocarbons in ocean water. 1. Instrumentation and background correction. Analytical Chemistry, 68, 3534-3538.
  A spectrofluorometer capable of dispersed-spectrum, simultaneous, multiwavelength UV excitation and collection of luminescence has been constructed for the purpose of qualitatively and quantitatively determining aromatic hydrocarbon pollutants dissolved in ocean water. Hydrocarbon fluorescence data produced by this instrument were in the form of excitation/emission matrices, which provide more spectral information about these complex mixtures than is available from conventional excitation or emission fluorescence profiles. Second-order statistical methods were applied to these data to determine low part-per-billion concentrations of two primary fluorescent compounds, naphthalene and styrene, found in ocean water exposed to gasoline despite the presence of uncalibrated interference from similar aromatic compounds, the ocean water matrix, and the instrumental background.

• Muthén, B., Olsson, U., Pettersson, T. & Stahlberg, G. (1977).
  Measuring religious attitudes using the semantic differential technique: An application of three-mode factor analysis. Journal for the Scientific Study of Religion, 16, 275-288.
  120 theology students were presented with 6 religious concepts to be scored on 60 semantic bipolar seven-point scales. These data were subjected to T3. A straight forward application and interpretation.

           

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