Three-mode component models: A review of the literature

(1)

THREE-MODE COMPONENT MODELS

A SURVEY OF THE LITERATURE

Pieter M. Kroonenberg

Department of Education, Leiden University

This paper is a part of a larger review paper on three-way techniques. In particular, component models are reviewed, with special emphasis on PARAFAC and the Tucker models.

1. INTRODUCTION

This paper is an attempt to sketch the major areas of development in component methods which are explicitly designed to handle three-way data which have three different modes, such as profile data. This may be contrasted with data which have the shape of sets of similarity matrices and sets of covariance matrices, which are two-mode three-way data. Carroll and Arabie ( 1980) provide a thorough discussion of the differences between these data types. The emphasis is on methods and algorithms for three-way data, which deal with the three-way structure in an explicit manner, and which are primarily model-based. This eliminates two-way methods which work on strung-out (or unfolded, as the chemometricians have it) matrices, and those techniques which concatenate several different techniques rather than presenting one single frame of reference. Furthermore, primarily mathematical aspects will not be discussed, such as the rank of three-way matrices (see e.g. Kruskal, 1976, 1977, 1989; Franc, 1989; Ten Berge, Kiers, & De Leeuw, 1988; Ten Berge, 1991), and optimality properties (see d'Aubigny& Polit, 1989). Nor will specific attention be paid to n-way extensions to the methods discussed here.

2. COMPONENT MODELS

(2)

620 PM. Kroonenberg

estimate unique factors or unique variances.

Three-way multivariate analysis started with the seminal work of Ledyard Tucker (especially, 1966), and much work in this area is in one way or another derived from his basic ideas. We will look at his principal component models (the TuckerS and Tucker2 models) and the so-called PArallel FACtor model - PARAFAC (Harshman, 1970, 1972; Harshman & Lundy, 1984a,b), which is also known, primarily in the field of multidimensional scaling as the CANcnical DECOMposition model - CANDECOMP (Carroll & Chang, 1970).

In an interesting set of papers, Kiers (1988,1992) shows how several models which are to be discussed here can be arranged into hierarchies of component models with increasing numbers of restrictions on the model parameters. Kiers (1988) does this for the Tucker2 family and Kiers (in press- a) for the Tucker3 family. Geladi (1989) is a tutorial for the methods with special reference to chemical applications. Other papers which discuss relationships between models in this area are Lohmöller (1979), Carroll & Arabie (1980), Kroonenberg (1983a, chap. 2; 1983b, 1988), Snyder, Law, & Hattie (1984), Harshman and Lundy (1984a), Leurgans and Ross (1991), and Smilde (1991).

2.1 PARAFAC MODEL

As with most of the models in this section, the PARAFAC model is one of the generalisations of the singular value decomposition to three-way data. Several theoretical discussions about the nature of such generalisations can be found in Denis and Dhorne (1988), Franc (1988), Yoshizawa (1987), and Kroonenberg (1989).

The PARAFAC model (Harshman, 1970) for a three-way array X with elements x|]k has the form

s

X,|k = Za lsb|SCks9ssS + e,jk (1)

s 1

where ais, b|S, and cks are the elements of the component matrices A, B, and C, respectively, and gsss is the element of a superdiagonal core cube G (i.e. gpqr = 0 if p* q* r), and e . are the errors of approximation. Another common notation for this model using Kronecker product is the so-called tensor form

s

X = £ as® bs® cs+ E ( 2 )

s 1

(3)

of) authors, Carroll and Chang (1970; using the name CAnonical DECOMposition) and Harshman (1970, 1972; using the name PARAIIel FACtor analysis). To date, Harshman & Lundy (1984a,b) is the most complete treatment of the model and many related issues. The model has been subject to considerable research in various disciplines.

Models such as PARAFAC are often called TRILINEAR, because they are linear in each of the component matrices given the other two. Similarly, the TuckerS model is a QUADRILINEAR model (see e.g. Kruskal, 1984).

2.1.1 UNIQUENESS

Under fairly general identification conditions, both proposers have shown the model to have a unique solution, in other words the model is identified, and it is especially this feature which has attracted much attention (see also Leurgans and Ross, 1991, for a detailed mathematical treatment). Whereas two-way exploratory factor analysis suffers from rotational indeterminacy, i.e. the solution may be non-singularly transformed without affecting the quality of the fit of the model, such indeterminacy disappears in the three-way case due to the identifiability. Of course, having a restricted model implies that there are situations where the model does not fit. The present iterative algorithm seems to be somewhat sensitive to local minima, so that one needs several runs from different starting positions to evaluate the 'best' minimum. Harshman and Lundy (1984a,b; Harshman & DeSarbo, 1984) discuss at length the possibilities and problems of analysing three-way data with the PARAFAC model.

Harshman uses the uniqueness property (or 'intrinsic axis property') to search for "real" psychological factors. As the parameters depend only on one of the indices i, j, k, they may be seen as proportionality constants for entire factors. For example, the factor as is proportionally enlarged or decreased in a level k of the third way by an amount cks. Thus across conditions the factors as are parallel. Hence the name of the model. Harshman's basic claim is that only "real" factors will behave in such a way, and if the model fits, the factors should be due to some interesting psychological phenomenon.

(4)

622 P.M. Kroonenberg

of'generalized rank annihilation') is used more as a model for parameter estimation than for discovery. Some relevant papers in this area are the review papers by Geladi (1989) and Smiide (1991 ), the exposition by Leurgans and Ross (1991 ), and papers by Sanchez and Kowalski (1988,1990) and Appellof and Davidson (1981). The latter authors, in fact, propose to use Tucker's (1966) Method I analysis (without explicitly mentioning this) for reducing a data set to a more manageable size, and propose to follow on with a PARAFAC analysis on the resulting core matrix.

2.1.2 ALGORITHMS

The basic algorithm for the PARAFAC model is based on an alternating least squares (ALS) approach, i.e. the parameters in the model are divided into groups which can be estimated succesively holding values of the other groups fixed. The iteration procedure cycles through the substeps until convergence is reached. Leurgans and Ross (1991, p. 20) provide a simple proof that the components cannot be found one by one (or recursively) unless orthogonality conditions are imposed, because PARAFAC models with fewer terms than are actually present cannot recover the parameters of larger models. Within each step of a major iteration in the simultaneous procedure, three regression problems are solved, each for one of the ways with the other two ways fixed. As reported by most authors convergence is rather slow and alternative routes have been investigated. Appellof and Davidson (1981) reported testing a Fletcher-Powell algorithm (Fletcher & Powell, 1964) which was anything but an improvement. They also investigated the performance of the so-called Aitken extrapolation method which worked reasonably well in accelerating convergence. Harshman uses in his program a similar "overrelaxation method" (Harshman, 1970, p. 33).

Kiers and Krijnen (1991) suggest a method for improving the basic algorithm for the PARAFAC model by estimating the parameters using variable-by- condition covariance matrices rather than the raw profile data. They report considerable decreases in computer time for random data. It was shown that at each iteration step the loss function is the same as that of the original algorithm. As an interesting sideline, it should be noted that the modified PARAFAC algorithm does not use the raw data but only the (means and) covariances for the basic computations, so that for this three-way method, as for their two-way counterparts, the first and second order moments are sufficient statistics.

(5)

an algorithm based on estimating succesive differences in each iteration and solving sets of simultaneous equations to find the required parameters Hayashi and Hayashi (1982) reported identical results to those of Harshman's PARAFAC program.

2.1.3 ALGEBRAIC SOLUTION

Sands and Young (1980), using a suggestion by Yoshio Takane, presented a decomposition which will provide a solution to the PARAFAC model in the case of a perfect fit. The basic idea of this decomposition goes back to Schönemann ( 1972), who proposed an algebraic solution for the scalar-product version of Carroll and Chang's (1970) INDSCAL model. Leurgans, Ross, and Abel (1990), Sanchez and Kowalski (1990) and Burdick, Tu, McGown, and Millican( 1990) independently developed a comparable procedure to Sands and Young with minor variations (for a comparison see Leurgans et al., 1990). In the case of approximate solutions, the Sands & Young procedure and its variants can be used as an initialization for an alternating least squares algorithm, in the same manner as the Tucker's (1966) Method I is used as initialization of the alternating least squares algorithms for the Tucker models (see below).

2.1.4 OPTIMAL SCALING

Sands and Young (1980) also developed an optimal scaling version of PARAFAC (ALSCOMP3) within the optimal scaling tradition of Takane, Young, & De Leeuw (e.g., 1977). The inclusion of an optimal scaling phase allows for the treatment of data with different measurement levels, and for data with row, column, or matrix conditionalities (see e.g. Gifi, 1990, for a discussion of these terms). Harshman and Lundy (1984a, p. 188 ff.) give their opinion on the relative merits of Sands and Young's ALSCOMP3 procedure and PARAFAC.

2.1.5 CONSTRAINTS

Another aspect of the PARAFAC model is the possibility of imposing constraints on the parameters. Much work in this area has been done by Carroll and co-authors (Carroll, Pruzansky, & Kruskal, 1980; Carroll & Pruzansky, 1984; Carroll, De Soete, & Pruzansky, 1989), see, for a detailed mathematical treatment: Franc (1992).

(6)

624 " PM Kroonenberg

DEcomposition with LINear Constraints). The theorem due to Kruskal presented in this paper shows that a similar procedure could be incorporated in the alternating least squares solutions of the Tucker3 and Tucker2 models as well (see below).

Another innovation is the use of the PARAFAC model for fitting Lazarsfeld and Henry's (1968) latent class model for contingency tables (Carroll, De Soete, & Pruzansky, 1989). The latent class model specifies that all observed dependencies or interactions in the contingency table are caused by the fact that people belong to different latent classes. Or given the latent classes, the variables making up the table are independent. The model has the same form as the PARAFAC model, be it that probabilities require non-negativity constraints. Carroll, De Soete, and Pruzansky (1989) developed an algorithm using iterative reweighted least squares to produce best asymptotically normal estimates for the parameters, which incorporated the non-negativity requirements at the same time.

One may also put orthogonality restrictions on PARAFAC components. Harshman and Lundy(1984a;Harshman,l_undy, & Kruskal, 1985;Lundy, Harshman, & Kruskal, 1989) use such restrictions on configurations to prevent certain kinds of degenerate solutions (see also, Kruskal, Harshman, &Lundy, 1989). The inclusion of orthogonality constraints in Harshman's PARAFAC program are available, but not publicly documented. Kiers (1988, 1991) used such orthogonal PARAFAC models in his hierarchies of component models for three-way data.

Harshman, Lundy, and Kruskal (1985), Lundy, Harshman, and Kruskal (1989), and Kruskal, Harshman, and Lundy (1989) discuss a procedure, PFCORE, which relaxes constraints on the solution. After deriving an orthogonally constraint PARAFAC solution they suggest using the results from that solution to investigate the nature of the core matrix for a Tucker3 model (see below). It employs a version of formula (4), using generalised inverses rather than transpositions of the component matrices. This allows them to evaluate the nature of the degeneracy of an unrestricted PARAFAC solution.

Durrell, Lee, Ross, and Gross (1990) refer to programs for three-way and four-way PARAFAC models (Lee, 1988) which also include nonnegativity constraints, and also Carroll, De Soete, and Pruzansky (1989) discuss the inclusion of nonnegativity constraints.

2.2 TUCKER3 MODEL

(7)

p Q R

Xi | k = Z Z Za,pb)qckr9pqr+e1|k (3)

p 1 q 1 r 1

where a , b and ckr are the elements of the component matrices A, B, and C, respectively, and g is an element of the three-way core matrix G

Tucker's (1966) method to estimate the parameters was based on the stringing-out (or unfolding) of the data matrix in three different ways, and performing a principal component analysis on each of them giving A, B, and C, respectively. The core matrix G is then computed from the three component matrices as

l J K

9 p q r = L I SXV*** (4)

l 1 | 1 k 1

(8)

626 PM Kroonenberg

only making iteration steps for the component matrices A, B, and C, but also for each slice Gr (r=1 ,..,R) of the core matrix G At the same time, they showed how missing data procedures could be incorporated in the algorithm. An as yet unexplored extension of this algorithm was also indicated by Weesie and Van Houwelingen Rather than using basic least squares regression, it should be possible to use robust regression, such as ridge regression, to prevent undue influence of outlying observations. Verhees (1989) sketches the theory of maximum likelihood estimation for the Tucker3 model, and gives several statistical details as well.

Building on their ideas, Kroonenberg has now included a missing data procedure in the Kroonenberg et al. (1989) version of the TUCKALS3 algorithm, using one more ALS-step to estimate the missing data from the model. As such an approach requires the re-estimation of missing data points at each step of the algorithm, it is not possible to include this method of handling missing data in the Kiers et al. (in press) version as well.

Compared to PARAFAC much less research has been carried out to incorpo-rate constraints into the estimation procedure for the Tucker3 model. To estimate the parameters of the model, Kroonenberg and De Leeuw (1980) put orthogonality constraints on the three component matrices. This can be done without loss of generality as after a solution is obtained the component matrices may be transformed nonsingularly without loss of fit. Much more interesting is a search for solutions with restrictions on the core matrix. If these can increase the interpretability of the core matrix, this would be of great help. Kiers (1992) discusses several proposals and algorithms for constraining the core matrix to be superdiagonal. In that case the Tucker3 model is equivalent to the PARAFAC model provided there are no orthogonality constraints on the components (see also Carroll & Chang, 1970, p.302). In addition, Kiers looked at the situation where specific elements of the core matrix are also set to zero, in which case the minimization has to procède with four multiple regression substeps in each major iteration.

(9)

2.3 TUCKER2 MODEL

The Tucker2 model (Jennrich, 1972; Kroonenberg & De Leeuw, 1980) for a three-way array X with elements x|jk has the form

P Q

X. j k = Z IXbjqhpqk + 6,jk (5)

P 1

where a and b are the elements of the component matrices A and B, respectively, and h k is an element of the three-way extended core matrix H.

The solution of the Tucker2 model is essentially similar to that of the Tucker3 model and and will not be discussed here. Also the improvements suggested for the TuckerS model carry over in a natural way. Murakami (1983) developed a variant of the TUCKALS2 algorithm resembling the Kiers, Kroonenberg, and Ten Berge (in press) improvements for the TuckerS model. Apart from the differences in the models the major difference is that Murakami uses eigendecompositions, while Kiers et al. use Gram-Schmidt orthonormalisations. Apart from Murakami, there seem to be few or no authors who have explicitly investigated alternative methods for estimating the parameters in the Tucker2 model. Harshman (1972; Harshman and Lundy, 1984a) suggest several models intermediate between the Tucker2 model and PARAFAC but they will not be discussed here, as their characteristics are still very much under investigation.

The situation for the Tucker2 model with respect to orthogonality of the components, and the inclusion of linear constraints is essentially similar to that of the TuckerS model, so that they need not be elaborated here. Far more interesting is to consider here constraints on the extended core matrix. As has been shown by several authors (Tucker, 1972; Kroonenberg & De Leeuw, 1980; Kroonenberg, 1983a, chap. 2; Harshman & Lundy, 1984a), demanding that the extended core matrix is diagonal amounts to specifying the PARAFAC model provided the orthogonality restrictions on the components are dropped.

(10)

628 . P.M. Kroonenberg

Kroonenberg showed that if a perfect PARAFAC exists a TUCKALS2 followed by a PARAFAC analysis on the core will provide the same solution.

2.4 COMMENTARY

As mentioned at several places there are various links between the methods discussed. It is not our intention to go into them here in many more detail than has been done so far. Such relationships often rest on mathematical similarities, but also (unsystematic) practical investigations have been carried out by various authors (see above). It would, however, be helpful if such investigations could be done in a more systematic manner, so that the relative merits can be assessed for those situation in which several models compete in their applicability.

3. COMMENTARY AND PERSPECTIVES

In the above sections, the general focus has been on models and algorithms, but there are several issues in connection with these models which have not been mentioned so far. Very prominent, for instance, in Harshman's work on developing the PARAFAC model, has been the question of preprocessing (i.e. centring and standardisation) of the data before a three-way analysis proper. Harshman and Lundy (1984b) discuss this issue in great detail touching on both algebraic and practical aspects (see also Kroonenberg, 1983a, chap. 6). Ten Berge and Kiers (1989) and Ten Berge (1989) provide some theoretical results with respect to the iterative centring and standardisation proposed by Harshman and Lundy.

Another issue in this context is the postprocessing of output, i.e. representation, graphing, and transformations of the basic output of the programs to enhance interpretability (see especially Harshman & Lundy, 1984b, and Kroonenberg, 1983a, chap. 6).

Smilde (1991) raises the issue of variable selection for three-way data, as well as the problem of non-linearities in the data and their effect on the solutions. These issues can also be seen as a serious concern in such areas like ecology where nonlinearities are rule rather than exception (see e.g. Faith, Minchin, & Belbin, 1987).

(11)

As is evident from the above the three-mode component models have undergone considerable development since Tucker set about to explore them. Commonly, technical development takes its lead from needs felt in applied fields, it soars ahead gaining its own momentum and in many ways leaves the substantive field far behind, notwithstanding its efforts to keep up. Characteristic in such cases is that few applications appear, and those that appear generally have been realised in close cooperation with the originators of the method.

The availability of computer programs to carry out the analyses, accompanied by adequate documentation, is clearly a conditio sine qua non for a widespread usage of statistical methods. In a way, the programs and manuals are a substitute for the originator as personal assistant. Such independence from the originators should of course be strongly stimulated, and its occurrence is a sign of coming of age. In that sense, one might say that certain areas in the three-way world certainly have come of age.

REFERENCES

Appellof. C J . & Da vidson, E R . 1981. Strategies for analyzing data from video fluorometnc monitoring of liquid Chromatographie effluents Analytical Chemistry. 53. 2053-2056

Brouwer. P .&Kroonenberg. P.M., 1991. Some notes on the diagonalization of extended core matrices

Journal of Classification. 8. 93-98.

Burdick. D.S , Tu, X M . McGown. L B . & Millican. D.W.. 1990. Resolution of multicomponent fluorescent mixtures by analysis of the excitation-emission-frequency array Journal of

Che-mometrics. 4. 15-28.

Carroll. J D . & Arabie. P . '\98Q.Multidimensionalscaling Annual Review of Psychology. 31.607-649 Carroll. J D . & Chang. J.J.. 1970. Analysis of individual differences in multidimensional scaling via an

N-way generalization of "Eckart-Young" decomposition Psychometrika. 35. 283-319

Carroll, J D , De Soete. G . & Pruzansky. S . 1989. Fitting of the latent class model via iteratively reweighted least squares candecomp with nonnegativity constraints In R Coppi and S Bolasco (Eds.). Multiway data analysis Elsevier. Amsterdam, (pp. 123-130)

Carroll. J.D.. & Pruzansky. S , 1984. The CANDECOMP-CANDELINC family of models and methods for multidimensional data analysis. In H.G Law. C W Snyder Jr.. J.A Hattie. & R P McDonald (Eds.). Research methods for multimode data analysis Praeger. New York, pp 372-402 Carroll. J.D.. Pruzansky. S . & Kruskal J.B., 1980. Candelinc: A general approach to multidimensional

analysis of many-way arrays with linear constraints on parameters Psychometrika. 45. 3-24. Carroll, J.D., & Wish. M . 1974. Models and methods for three-way multidimensional scaling. In D.H

Krantz. R C Atkinson. R.D Luce. & P Suppes (Eds ) Contemporary developments in mathematical psychology Freeman, San Francisco. Vol II pp. 57-105

Cohen, H.S.. 1974. Three-mode rotation to indscalstructure (TRIAS) Paper presented at Psychometric

(12)

D Aubigny, G , & Polit, E , 1989, Some optimally propoerties of the generalization of n-way tables with

specified matrices In R Coppi and S Bolasco(Eds.), Multiway data analysis Eisevier, Amsterdam , pp 39-52

De Ligny, C , Spanjer. M., Van Houwelmgen, H . & Weesie, J , 1984, Three-mode factor-analysis of

data on retention in normal-phase high-performance liquid chromatography Journal of Chromatography, 301, 311-324

Denis, J B , & Dhorne, T . 1989, Orthogonal tensor decomposition of 3-way tables In R Coppi and S. Bolasco (Eds ) Multiway data analysis Elsevier, Amsterdam, pp. 31- 38

Durrell, S.R., Lee, C.-H., Ross, R.T., & Gross, E.L., 1990, Factor analysis of the near-ultraviolet absorption spectrum of plastocyanin using bilnear, trilinear, and quadrilinear models Archives of Biochemistry and Biophysics, 278, 148-160

Faith, D P , Minchin, P , & Beibin, L 1987, Compositional dissimilarity as a robust measure of ecological

distance Vegetatio, 69, 57-68

Fletcher, R , & Powell M J D , 1964. A rapidly convergent descent method for minimization The Computer Journal, 2, 163-168.

Franc, A , 1989. Multiway matrices: some algebraic results In R Coppi and S Bolasco (Eds ), Multiway data analysis Elsevier, Amsterdam, pp 19-30

Franc, A : Étude algébrique des multitableaux: Apports de l'algèbre tensorielle. Thèse de 3*m" cycle.

Université Montpellier II, France

Geladi, P., 1989, Analysis of multi-way (multi-mode) data Chemometrics and Intelligent Laboratory Systems?, 11-30.

Gifi, A , 1990, Nonlinear multivanate analysis Wiley, Chicester

Gower, J.C.. 1985, Generalised Procrustes analysis Psychometrika, 40, 33-51

Harshman, R.A, 1970, Foundations of the PARAFAC procedure: Models and methods for an "explanatory" multi-mode factor analysis UCLA Working Papers in Phonetics, 16, 1-84 Harshman, R A , 1972, Determination and proof of minimum uniqueness conditions for PARAFAC1

UCLA Working Papers in Phonetics, 22, 111-117

Harshman, R A , & Lundy, M E , 1984a, The PARAFAC model for three-way factor analysis and multidimensional scaling In H.G Law, C W Snyder Jr., J A Hattie, & R P McDonald (Eds ), Research methods for multimode data analysis Praeger, New York pp 122-215

Harshman, R A ,& Lundy, M E ,1984b, Data preprocessing and the extended PARAFAC model. In H.G Law, C W Snyder J r , J A Hattie, & R P. McDonald (Eds ), Research methods for multimode data analysis Praeger, New York, pp 216-284.

Harshman, R A., Lundy, M E , & Kruskal, J.B., 1985, Comparison of trilinear and quadrilinear methods:

Strengths, weaknesses, and degeneracies Paper presented at the annual meeting of the Classification Society, St.John, Newfoundland. Canada.

Harshman, R A., & DeSarbo, W.S., 1984, In H G. Law, C.W. Snyder J r , J A Hattie, & R.P McDonald (Eds.), Research methods for multimode data analysis Praeger, New York, pp 602-642 Hayashi.C &Hayashi, F , 1982. A new algorithm to solve PARAFAC-model Behaviormetrika,

11,49-60

Hayashi, F , Yamaoka. K &Terao, H . 1982, A comparative study between two algorithmsofPARAFAC model (in Japanese) Japanese Journal of Behaviormetrika. 1, 47-62

Jaffrenou, P.A., 1978, Sur l'analyse des familles finies de variables vectorielles On the analysis of finite

(13)

Jennrich, R., 1972, A generalization ol'the multidimensional scaling model of Carroll and Chang UCLA Working Papers in Phonetics, 22, 45-47

Kiers, HAL, 1988, Comparison of "Anglo-Saxon" and "French" three-mode methods Statistique et Analyse des Donn es, 13, 14-32.

Kiers, H.A.L., 1989a, INDSCAL for the analysis of categorical data In R Coppi and S Bolasco (Eds.), Multiway data analysis Elsevier, Amsterdam, pp. 155-167.

Kiers, H.A L, 1989b, Three-way methods for the analysis of qualitative and quantitative two-way data DSWO Press, Leiden

Kiers, H A.L., 1991, Hierarchical relations among three-way methods Psychometrika, 56, 449-470

Kiers, H A L : TUCKALS core rotations and constrained TUCKALS modelling, Statistica Applicata, n° 3, Vol 4

Kiers, H A L . , & Krijnen, W.P., 1991, An efficient algorithm for PARAFAC of three-way data with large numbers of observation units Psychometrika, 56, 147-152

Kiers, H A L , Kroonenberg, P.M.. & Ten Berge, J. M. F.: In press, An efficient algorithm for TUCKALS3 on data with large numbers of observation units. Psychometrika

Kroonenberg. P.M., 1983a, Three-mode principal component analysis: Theory and applications DSWO Press Leiden

Kroonenberg, P M , 19836, Annotated bibliography of three-mode factor analysis British Journal of Mathematical and Statistical Psychology, 36, 81-113.

Kroonenberg, P.M., 1988, Three-mode analysis In S Kotz & N L Johnson (Eds.), Encyclopedia of Statistical Sciences Wiley, New York, Vol 9, pp. 231-236.

Kroonenberg, P.M., & De Leeuw, J , 1980, Principal component analysis of three-mode data by means of alternating least-squares algorithms Psychometrika, 45. 69-97.

Kroonenberg, P.M., Ten Berge, J.M.F., Brouwer, P., & Kiers, H A L , 1989, Gram-Schmidt versus Bauer-Rutishauser in alternating least-squares algorithms for three-mode principal component analysis. Computational Statistics Quarterly, 5, 81-87

Kruskal, J.B., 1976, More factors than subjects, tests than treatments: An indeterminacy theorem for canonical decomposition and individual differences scaling Psychometrika, 41, 281-293

Kruskal, J.B.. 1977, Three-way arrays: Rank and uniqueness of tnlinear decompositions, with application to arithmetic complexity and statistics Linear algebra and its applications, 18, 91-138 Kruskal, J.B., 1984, Multilinear methods In H G Law, C.W Snyder J r , J A Hattie, & R.P. McDonald

(Eds.), Research methods for multimode data analysis Praeger, New York. pp. 36-72

Kruskal, J.B., 1989,»Ran/<, decomposition, and uniqueness for 3-way and N-way arrays In R. Coppi and S Bolasco (Eds.), Multiway data analysis Elsevier, Amsterdam, pp 7- 18

Kruskal, J.B., Harshman, R.A.. & Lundy. M E , 1989. How 3-MFA data can cause degenerate PARAFAC solutions, among other relationships In R Coppi and S Bolasco (Eds.). Multiway data analysis Elsevier, Amsterdam, pp 123-130

Lazarsfeld, P.F., & Henry, N W : Latent structure analysis Houghton Mifflin. Boston

Lee, C.-H., 1988, Multilinear analysis of fluorescence spectra of photosynthetic systems PhD. Thesis. The Ohio State University, Columbus, OH.

Leurgans. S E , & Ross, R.T.: In press Multilinear models: Application in spectroscopy Statistical

(14)

Leurgans. S E , Ross. R T . & Abel. R.B., 1990, A decomposition for 3-way arrays. Technical Report

No 448 Departments of Statistics and Biochemistry Ohio State University, Columbus, OH. Lohmöller, J.-B , 1979, Die trimodale Faktorenanalyse von Tucker: Skalierungen, Rotationen, andere

Modelle Tucker's three-mode factor analysis: Scaling, rotations, other models Archiv für die

Psychologie. 131, 137-166

Lundy, M.E., Harshman, R A , & Kruskal, J.B., 1989, A two-stage procedure incorporating good features of both tn/inear and quadri/inear models. In R Coppi and S Bolasco (Eds.), Multiway data analysis Elsevier, Amsterdam, pp 123-130

Murakami, T , 1983, Quasi three-mode principal component analysis - a method for assessing the factor change Behaviormetrika, 14, 27-48

Sanchez, E , & Kowalski, B R , 1988, Tensorial calibration II Second-order calibration Journal of Chemometncs, 2. 265-280

Sanchez, E , & Kowalski, B R , 1990, Tensorial resolution: A direct trilinear decomposition Journal of Chemometncs. 4, 29-45

Sands, R , & Young, F.W., 1980, Component models for three-way data: ALSCOMP3. an alternating least squares algorithm with optimal scaling features Psychometnka, 45, 39-67

Schönemann, P H , 1972, An algebraic solution for a class of subjective metric models Psychometrika,

37,441-451

Smilde.A K : Three-way analyses: Problems and perspectives Chemometncs and Intelligent Laboratory Systems (in press)

Snyder J r , C.W., Law, H.G., & Hattie, J A , 1984, Overview of multimode analytic methods In H G Law, C W Snyder J r , J A Hattie, & R P McDonald (Eds.). Research methods for multimode data analysis Praeger, New York, pp 2-35.

Spanjer, M , 1984. Substituent interaction effects and mathematical-statistical description of retention in liquid chromatography Ph D Thesis, University of Utrecht

Takane.Y , Young.F W , & De Leeuw, J , 1977, Nonmetric individual difference multidimensional

scaling An alternating least squares method with optimal scaling features Psychometrika,42,7-67. Ten Berge, J M F , 1989, Convergence of PARAFAC preprocessing procedures and the

Demmg-Stephan method of iterative proportional fitting In R Coppi and S Bolasco (Eds ), Multiway data analysis Eisevier, Amsterdam, pp 53-64

Ten Berge, J.M.F., 1991, Kruskal's polynomial for 2x2x2 arrays and a generalization to 2xnxn arrays. Psychometrika, 56, 631-636.

Ten Berge, J.M.F , De Leeuw. J , & Kroonenberg, P M . 1987, Some additional results on prinicpal components analysis of three-mode data by means of alternating least squares. Psychometrika,

53, 183-191

Ten Berge, J M.F., AKiers, H A L , 1989, Convergence properties ofan iterative procedure of ipsatizing and standardizing a data matrix, with applications to Parafac/Candecomp preprocessing Psychometrika, 54, 231- 236;

Ten Berge, J MF., Kiers, HAL, A De Leeuw, J., 1988, Explicit candecomp/PARAFAC solutions fora

contrived 2x2x2 array of rank three Psychometrika, 53, 579-584

Tucker, L.R., 1966, Some mathematical notes on three-mode factor analysis.

(15)

Weesie, J , Van Houwelingen, J , 1983, GEPCAM user's manual Generalized principal components analysis with missing values Technical report, Institute of Mathematical Statistics, University of

Utrecht

Tucker. L.R., 1972. Relations between multidimensional scaling and three-mode factor analysis

Psychometnka, 37. 3-27

Yoshizawa, T , 1987. Singular value decomposition of multiarray data and its applications Technical

report CS-87-038, Department of Computer Science, Yamanashi University, Kofu, Japan

RIASSUNTO