(Reverse chronological order)

- Harshman,
R. A. Multimode methods, tensors, and 'truth'. Paper presented at IMPS
2007, the Sixteenth International Meeting of the
**Psychometric Society**, Tokyo, Japan, July, 2007. (invited) Parafac (PARAllel FACtor analysis) generalizes Factor/Component Analysis (FCA) by increasing the level of multilinearity. Certain useful properties that this creates have led, in the last decade, to a rapid expansion of Parafac applications in various areas of science and engineering. This talk briefly explains what some of these properties are, why they have been considered important, and then considers how increasing the multilinear level of other analysis methods can give them similarly useful added properties.**Summary**

I. Parafac raises the multilinear order of the factor/component model from bilinear to trilinear or n-linear. It fits data arrays with three (or more) subscripts, such as measurements of n cases on m variables on each of p occasions, or correlations among n variables for p different groups. This extends the scope of the model and integrates information from multiple sources to provide stronger statistical estimates, but its most important consequence is a potential for full model identification. With three-mode and higher-way models, it is possible to get essentially unique factor loading estimates simply by maximizing the model's fit to a dataset -- provided the factor variations have appropriate structure and are adequately distinct. This means that a mixture of outer-product patterns can be correctly resolved into the individual contributions generated by the original sources of variation, without 'rotational ambiguity'. This is why Parafac is increasingly used in applications where the 'truth' of the recovered factors is both essential and objectively verifiable, e.g., separation of the complex spectrum from a physical mixture of chemicals into the spectra of the individual compounds in the mixture, correct 'blind separation' of mixed telecommunications or radar signals, image analysis, etc.

II. Other statistical methods can be generalized in a similar way, and acquire similarly enhanced model identifiablity and pattern recovery. This has been demonstrated with canonical correlation and hence the GLM, and with some kinds of Structural Equation Modeling. Basically, a traditional method that finds an optimal linear combination across one index of a two-way data array (combining columns of data), can be generalized to one that finds a linear combination jointly-optimal across two (or more) indices of a three- (or higher)-way array. The figure below shows how the General Linear Model (left) is modified to increase the multilinear order of its source data (X) and canonical weights (W). This generalization implies corresponding generalization of the various special cases of the GLM, such as Discriminant Analysis, (M)ANOVA /(M)ANCOVA, etc. Significance tests can be performed by bootstrapping and permutation methods. As with Parafac, model identification is often made easier, and requires fewer artificial constraints. In the generalized canonical model, for example, axes corresponding to 'true' (empirical-source) patterns can sometimes be recovered uniquely in the left and right canonical spaces, and without orthogonality constraints. A further generalization of CC replaces the canonical vectors with canonical tensors of order 2 or higher. Not all issues have been resolved, however, and many of these methods are still under study and development.

III. Tensors and multilinear algebra form the natural basis for these methods. One way to visualize a tensor is as a long vector in a higher-dimensional 'product space' generated by interaction of the lower dimensional spaces in several modes. An alternative way is as a 'hyper-vector' whose components are themselves vectors or even tensors. While the components have multiple subscripts, the component values recombine under a change of basis just as do components or projections of a simple vector. Tensors can be powerful psychometric tools for compact description of psychological properties or processes that have richer behavior than single vectors; just as they now represent multi-vector physical properties like stress inside a twisted medium or elasticity.

IV. All these multilinear statistical models (e.g., Parafac, multilinear SEM or multilinear regression) can be considered neutral with respect to the distinction between "component-like" analysis, which considers all the variance, and "common-factor-like" analysis, which considers only variance shared among variables. A new method of array "filtering" has been developed which allows any model that is directly fit to a data array to (optionally) be fit only to its common-factor variance. - Harshman, R. A. Two successful and two
problematic applications of Parallel Factor Analysis (Parafac) in
Psychology. Paper presented at IMPS 2007, the Sixteenth International
Meeting of the
**Psychometric Society**, Tokyo, Japan, July, 2007. (invited) Two successful applications show how Parallel Factor Analysis (Parafac) can provide information not obtainable with FA/PCA. Then, two less successful applications show that Parafac is not always appropriate and requires more careful and insightful use than PCA.**Summary**

Example 1, Confirmatory-Exploratory FA: In an earlier study of the cross-cultural generality of personality dimensions, the Jackson PRF and Paunonen's NPQ were administered in 8+ countries and their two-way factor solutions were compared. A reanalysis using Parafac and Parafac2 obtained a single set of factors for all eight countries plus differential weights for each country. These factors are unique (fully identified) without rotation. In the 5D and 6D solutions, some resemble members of the 'Big 5' while others do not but are fully interpretible.

Example 2, A true Experiment. To resolve a longstanding debate about the axes of 2D 'emotion space', D. Stanley showed participants 25 movie scenes, selected to elicit various emotions, and after each scene participants rated their mood on 32 scales.These ratings were analyzed by Parafac to determine the directions that emotion space was stretched and contracted by the films. The results supported 'Valence-Arousal' against 'NA-PA' with high statistical significance (based on bootstrapping).

Problem Examples: Analyses of semantic differential and TV ratings show how Parafac behaves when it is inappropriate. Other Examples: The broad usefulness of Parafac and Parafac2 is indicated by numerous applications in Chemometrics, signal analysis (telecommunication and radar), brain imaging, phonetics, and elsewhere. - Harshman,
R. A., & Lundy, M. E. A randomization method of obtaining valid
p-values for model changes selected "post hoc". Poster presented at the
Seventy-first Annual Meeting of the
**Psychometric Society**, Montreal, Canada, June, 2006.View .pdf file (99 KB, 11 pp) When model changes are guided by**Summary***post hoc*assessment of the observed improvements in prediction or fit (e.g., in stepwise regression), it is usually impossible to obtain p-values for these improvements by conventional analytical methods. We describe a computer-intensive alternative that accurately estimates these p-values by using a modified randomization/permutation test procedure that empirically determines the appropriate null distributions. To demonstrate the method, we use it to get valid p-values for step inprovements found during standard stepwise multiple regression. Our method corrects for bias caused by the increased 'capitalization on chance' intrinsic to post hoc variable selection; it does this by introducing an equivalent post hoc selection step into the process generating the null-hypothesis values. The method also corrects for an "inconsistency" bias by eliminating or "pruning out" permutated cases that are inconsistent with prior step results; without such pruning, the method would underestimate significance except on the first step. In a Monte Carlo sample of one million cases, the p-values estimated for fit improvements during a three-step stepwise multiple regression did not show a statistically detectable bias at any step. Potential applications include significance tests for more complex sequential methods, stepwise canonical correlation/MANOVA, and discriminant analysis. - Harshman, R. A. Generalization of Parafac and Tucker models to canonical correlation models. Paper presented at
**TRICAP 2006**, a conference on three-way methods in chemistry and psychology, Crete, Greece, 2006. Tucker and Parafac are methods of �internal relationship analysis�: they find patterns of relations within a single dataset. However, statistical data analysis also uses �external relationship analysis�. External analysis methods find relationships between patterns in one dataset and those in another, or between patterns in a dataset and those in a theoretical logical structure or ``design matrix''. Bro and others have introduced external analysis into the three-way domain by deriving n-way extensions of PLS regression. I have stumbled upon an alternative approach, one that extends the symmetric relation of (canonical) correlation. Canonical correlation finds weights for an optimal linear combination of the columns of one matrix, and weights for an optimal linear combination of the columns of another matrix (the �other side� of the relation) such that the two vectors they define have maximal correlation. This is then repeated for vectors orthogonal to the first set, to get added dimensions of correlation between the two matrices. In statistics, canonical correlation is used to define the General Linear Model (GLM). By choice of arguments supplied to the GLM, one can obtain special cases that encompass a wide range of standard statistical techniques, including Univariate and Multivariate Analysis of Variance and Covariance, Discriminant Analysis, Multiple Regression, analysis of counts and contingency tables, and even the simplest �special case�, group comparisons via the t-test. I will describe an �external analysis� analog of Tucker T3, called TUCCON, and a Parafac-like special case of it called PARACCON, which provide multilinear generalizations of canonical correlation and thus of the GLM. This directly leads to multi-way versions of all the GLM�s ``special-cases'', often providing the added benefits for these methods that multiway generalization gives to Factor Analysis. For example: when, on either side of the canonical relation, there is a higher-way array in which the relevant patterns have a unique Parafac decomposition, then there also be a unique decomposition of the dimensions defining the �external relations'' between the two sides (i.e., of the �canonical space� on each side). Also, the source for patterns on each side of a canonical relation can now involve more than one dataset per side, allowing an outer-product synthesis of patterns from several sources on one side to be related to, say, the outer-product patterns in a single higher-way array on the other. Estimation of the parameters for TUCCON and PARACCON is often possible using familiar methods like Alternating Least Squares or Paatero�s Multilinear Engine. [ I will sometimes need to use the array notation AIN described in an earlier TRICAP meeting and published in the ``Index Formalism...'' article in Journal of Chemometrics. Interested colleagues are encouraged to gain some advance familiarity with this notation �downloadable from here. ]**Summary** - Bader, B. W., Harshman, R. A., &
Kolda, T. G. Improvements to three-way DEDICOM for applications in
social network analysis. Paper presented at
**TRICAP 2006**, a conference on three-way methods in chemistry and psychology, Crete, Greece, 2006. This presentation revisits the DEDICOM (DEcomposition into Directional COMponents) family of models and applies them to new applications in data mining, in particular social network analysis. The DEDICOM model is a tool developed in the late 1970's for analyzing asymmetric relationships in data analysis and has been revisited in the multiway community over the years. In this work we present an improved algorithm for computing the 3-way DEDICOM model, including modifications that make it possible to handle large, sparse data matrices. Then we will demonstrate the capabilities of DEDICOM as a new tool in social network analysis. For an application we consider the email communications of former Enron employees that were made public, and posted to the web, by the U.S. Federal Energy Regulatory Commission during its investigation of Enron. We represent the Enron email network as a directed graph with edges labeled by time and construct its corresponding adjacency tensor. Using the three-way DEDICOM model on this data, we show unique latent relationships that exist between types of employees and study their communication patterns over time.**Summary** - Harshman, R. A. Generalizing factor
analysis and canonical correlation to three-way arrays increases their
ability to disentangle information. Poster presented at the
Thirty-fourth Annual Meeting of the
**Statistical Society of Canada**, London, Canada, May, 2006.View .pdf file, no equations (53 KB, 4 pp) View .pdf file, with equations (69 KB, 3 pp) Data analysis is often improved when stronger models are applied to stronger data. We discuss one way of strengthening both: generalizing outer-product models from two-way (matrix) to three-way (array) form, and then applying them to datasets such as objects x variables x occasions, or multiple covariance matrices. Often, the three-way decomposition is 'essentially unique' without imposing orthogonality (or any other) constraints, eliminating rotational indeterminacies. Consequently, with appropriate data, one can recover, and/or relate across two arrays, approximations of the source patterns that originally generated the covariation, enabling new scientific applications (e.g., see Google: parafac). Three-way PCA/factor analysis by PARAFAC and canonical analysis by PARACCON (still under development) are discussed.**Summary** - Harshman, R. A., & Lundy, M. E.
Valid p-values for stepwise regression and other post-hoc model
selection methods. Poster presented at the Thirty-fourth Annual Meeting
of the
**Statistical Society of Canada**, London, Canada, May, 2006.View .pdf file (89 KB, 8 pp) When a model modification is selected using post-hoc information (e.g., in stepwise regression) it is hard to determine the statistical significance of the resulting improvement. We describe a computer-intensive method that empirically approximates the improvement's null distribution using permutation-test methods combined with "null-set pruning" (elimination of cases inconsistent with prior step results or other dataset characteristics). Monte Carlo tests indicate that un-pruned randomization gives unbiased p-values for post-hoc step 1, but biased ones thereafter; pruned randomization gives apparently unbiased p-values for successive steps as well (within the confidence bounds of our Monte Carlo test). Applications include complex sequential methods and stepwise canonical correlation/MANOVA.**Summary** - Harshman,
R. A. Multilinear generalization of the General Linear Model. Paper
presented at the Centre International de Rencontres Mathématiques
(CIRM)
**Workshop on Tensor Decompositions and Applications**, Marseilles, France, August, 2005. In statistics and data analysis there are two broad classes of methods: (i) ``internal relationship analysis'' methods like factor, component, and cluster analysis, which find patterns of relations within a single dataset and (ii) ``external relationship analysis'' methods that find relationships between patterns in one dataset and those in another, or between patterns in a dataset and those in a hypothesized logical structure or ``design matrix''. Canonical Correlation is the method used to test statements of ``external'' relations formulated in terms of the General Linear Model in statistics. By choice of arguments supplied to this method one can test ``special cases'' that encompass a wide range of standard statistical techniques including Univariate and Multivariate Analysis of Variance and Covariance, Discriminant Analysis, Multiple Regression, analysis of counts and contingency tables, and even simple group comparisons via the t-test. By generalizing Canonical Correlation from assessing linear relations between two matrices to assessing multilinear relations among N arrays of various orders, it becomes possible to formulate tensor-based generalizations of its many ``special-cases'', thereby open up new dimensions of ``external analysis''. We have recently seen how the extension of ``internal'' methods for factor and component analysis into higher orders by models like Parafac and Tucker can enrich and strengthen their capabilities. It now seems possible to achieve a considerably more fundamental and powerful expansion of statistical concepts and capabilities by extending the ``external analysis'' methods from linear to multilinear relationships. For example, multilinear regression can uncover more meaningful ``external relations'' that sort out dimensions of latent-structure in the linking patterns. Another multilinear method of regression uncovers relationships where Y is predicted from two or three different X matrices simultaneously, but each X is related to a different mode of Y. By incorporating the stronger uniqueness properties of some tensor decompositions, one sometimes improves parameter identifiably (e.g., in canonical correlation one can potentially eliminate the ``rotational'' ambiguity of orientation of axes in a canonical space). Because these extensions remain conditionally linear in subsets of their parameters, the structures they reveal are often relatively simple and/or can be broken into simple parts, and estimation of their parameters is often possible using familiar methods like Alternating Least Squares. [ I will sometimes need to use the simplified tensor notation described in the ``Index Formalism...'' article in Journal of Chemometrics, 2001, and interested colleagues are encouraged to gain some advance familiarity with this notation. The article can be downloaded from here. ]**Summary** - Harshman, R. A. Connections between the key properties of
tensors in physics and data analysis. Paper presented at the Centre
International de Rencontres Mathématiques (CIRM)
**Workshop on Tensor Decompositions and Applications**, Marseilles, France, August, 2005. A key property of tensors in Physics applications like Relativity is that a tensor describes an abstract entity or physical property in a way that is invariant with respect to changes in coordinates. That is, while the components of the tensor change the important abstract relations among them don't, one merely has different ``projections'' of the tensor-object onto a new reference frame. This gives a properly constructed equation using tensors a special status: it describes a general physical law or relation in a way that does not depend on a particular choice of one's reference frame (coordinate basis in each mode). Perhaps surprisingly, it is possible to carry over these principles into our use of tensors in data analysis, giving rise to interesting insights. In brief, our initial values of tensor components - the observed data - are in a reference frame that is determined by our measurement method(s) and circumstances. The tensor transformations we then apply during analysis change this reference frame to a new one in which the abstract tensor-object or objects cast ``shadows'' in the basis space of each mode that have improved properties such as simplicity or correspondence with causal source characteristics; these are hypothetical measurements that might have been obtained by alternative methods. Special transformations like PARAFAC focus on decomposable tensors, vectors in the tensor-product space that can be factored into vectors from the ``generating'' spaces in each mode - sometimes providing unique solutions. Our additive tensor decompositions are used to separate the combined ``shadows'' of a tensor sum into those of individual tensor objects. The key tensor property of pattern invariance and coordinate freedom not only justifies the use of basis transformations to simplify recovered patterns, but also could be important if we would use tensors to state mathematically more general theoretical conclusions (in Psychology, Chemistry, or whatever) that are not dependent on specifics of our original choice of measurement methods. Further parallels with physics suggest extensions of our current practice. For example, when our change of reference frames involve coordinate transformations that are nonorthogonal, they can create a dual-space relation among our reduced rank bases in different modes, making the distinction (until now, ignored) between ``covariant'' and ``contravariant'' tensors and indices potentially meaningful and useful in our work.**Summary** - Harshman, R. A. The Parafac model and its variants. Paper
presented at the Centre International de Rencontres Mathématiques
(CIRM)
**Workshop on Tensor Decompositions and Applications**, Marseilles, France, August, 2005. (invited)View .pdf file (251 KB, 52 pp) Parafac will be discussed both as a decomposition and as a model. From both perspectives, its simplicity and its potential for uniqueness are key properties. As a decomposition, it gives a minimal set of rank-1 components, similar to the SVD. As a model, this often turns out to be a good way to represent the pattern generated by a small number of source causes acting in n-way multimode data. For quite general mathematical/theoretical reasons (to be explained), the rank-1 outer-product form is often empirically meaningful. With Parafac, the implicit Tucker ``core'' is superdiagonal, and this independence of the component arrays simplifies their interpretation (though sometimes it is too restrictive). An even more important feature simplifying interpretation is the potential uniqueness of the decomposition; this is because Parafac uniqueness differs from that of the SVD or eigendecomposition. When certain conditions are met, Parafac can reveal data patterns that were actually generated by empirical sources (providing a method of ``blind recovery'') - if the uniqueness is determined by the latent sources of systematic variation (``deep uniqueness'') and not the noise or systematic error (``surface uniqueness''). Switching to a more applied perspective, data preprocessing and ways of testing convergence and assessing the correct number of factors when doing a Parafac analysis will also be covered. Next, some time will be devoted to the phenomenon of degenerate solutions, both what they are and how to avoid or overcome them. Some of the discussion will be a summary and extension of the geometric explanation given at the 2004 Tensor Decomposition Workshop (see that website). I will also propose a less worrisome interpretation of the fact that certain arrays do not have an exact best approximation at some lower ranks. Finally, variants and generalizations of Parafac will be considered ranging from Parafac indirect-fitting via covariances, Parafac2, PARALIND, PARATUCK2, and the Shifted Factor model, to alternatives and even more general multilinear and quasi-multilinear models. Differences in the properties and flexibility of these models (and/or decompositions) will be discussed. Finally, contrast will be made between n-way arrays that can be regarded as tensor-like and approximated by multilinear models like Parafac and Tucker, versus n-way arrays that should not be regarded as tensors because their systematic structure is known to be non-multilinear.**Summary** - Harshman, R. A. Generalization of canonical correlation to
N-way arrays. Paper presented at the Fourteenth International Meeting
of the
**Psychometric Society**, Tilburg, The Netherlands, July, 2005. - Harshman,
R. A. Harshman, R. A. The problem and nature of degenerate solutions or
decompositions of 3-way arrays. Paper presented at the American
Institute of Mathematics
**Tensor Decomposition Workshop**, Palo Alto, CA, July, 2004. (invited)View .pdf file (655 KB, 79 pp)

Supplementary information for the above paper:

Harshman, R. A. (2004). An annotated bibliography of articles on degenerate solutions or decompositions.View .pdf file (113 KB, 10 pp) - Harshman, R. A. Significance testing cuts both ways (or
should): Claims of "no real difference" should be supported at
p<.05. Paper presented at the Eighty-third Annual Meeting of the
**Transportation Research Board of the National Academy of Science (U. S.)**, Washington, D. C., January, 2004. (invited)View .pdf file (213 KB, 21 pp) - Harshman, R. A., & Hong, S. Multilinear models that include causal paths.Paper presented at
**TRICAP 2003**, a conference on three-way methods in chemistry and psychology, Lexington, Kentucky, June, 2003.Summary Familiar multilinear models, such as Tucker3 and PARAFAC, represent the simultaneous direct action of a few latent variables on many surface ones. Sometimes, it would be advantageous to have models that also allow the sequential (causal) action of some latent variables on others. This type of problem is currently addressed by means of Structural Equation Modeling, a method formulated in terms of covarince matrices. We present a different approach, based on direct fitting of N-way arrays by multilinear models that incorporate a path diagram or network. This approach can be used to extend existing models and/or to create new ones. - Harshman, R. A. A three-way generalization of chess. Poster presented at
**TRICAP 2003**, a conference on three-way methods in chemistry and psychology, Lexington, Kentucky, June, 2003.Summary Standard two-way chess is a game played by 2 people on a board of black and white squares with eight squares on each side. A three-way generalization is proposed, played by 3 people on a board of black, white, and red hexagonal cells, with eight cells on each player's side. At the start of the game, each player's first rank is the same as in two-way chess; it is protected, however, by two rows of pawns rather than one. (These are needed because of the greater vulnerability of pieces to attack on the hexagonal grid.) The rules of movement for the pieces are the same as in the two-way case, but applied to a hexagonally divided "playing space". If agreed upon at the start, each player also has a third bishop, to move along cells of the third color. The winning player is the one remaining after the others have been checkmated. (A further generalization to six-way chess is also described, which is identical except that it uses a bigger board and has a player on each of the six edges.) Hopefully, some game boards and sets of pieces will be available so that interested TRICAP participants can assess the generalization and, if desired, suggest alternative three-way models. - Lundy, M. E., Harshman, R. A., Paatero,
P., & Swartzman, L. C. Application of the 3-way DEDICOM model to
skew-symmetric data from paired preference ratings of treatments for
chronic back pain. Poster presented at
**TRICAP 2003**, a conference on three-way methods in chemistry and psychology, Lexington, Kentucky, June, 2003.Summary A 3-way generalization of the DEDICOM model (Decomposition into Directional COMponents) for skew-symmetric data (Harshman & Lundy, 1990) was applied to student paired preference ratings of 19 different treatments for chronic back pain. The model fitting process was accomplished using the Multilinear Engine program (Paatero, 1999). By imposing various constraints on the model during the data analysis, 3 distinct preference hierarchies amongst the treatments were identified: one amongst psychological treatments and herbal remedies, one amongst conventional medical treatments, and one amongst complementary/ alternative (CAM) physical treatments. The direction of preference within these hierarchies may be reversed for some people. Theoretical and practical implications are discussed.View .pdf file (142KB, 12 pp) - Swartzman,
L. C., Harshman, R. A., Lundy, M. E., & Burkell, J. Multivariate
approaches to assessing "implicit models" in the domain of health and
illness. Paper presented at the Twenty-third Annual Scientific Sessions
of the
**Society of Behavioral Medicine**, Washington, DC, April, 2002. - Swartzman, L. C., Harshman, R. A., Lundy,
M. E., Teasell, R. W., Burkell, J., & Chan, A. D. F. What are the
salient conceptual attributes of causes of chronic pain for
musculoskeletal pain patients? Poster presented at the Twenty-third
Annual Scientific Sessions of the
**Society of Behavioral Medicine**, Washington, DC, April, 2002. - Hong, S., & Harshman, R. A. Parafac2
fitting of covariance matrices from pooled and nested datasets with an
adjustment for factor mean differences. Paper presented at the
International Meeting of the
**Psychometric Society**, Osaka, Japan, July, 2001. - Swartzman, L. C., Harshman, R. A., Burkell,
J., Lundy, M. E., & Chan, A. D. F. What are the salient conceptual
attributes of causes of pain? Poster presented at the Twenty-second
Annual Scientific Sessions of the
**Society of Behavioral Medicine**, Seattle, Washington, March, 2001. - Harshman, R. A., Lundy, M. E., & Hong, S. Some
contributions to the search for identifiable models in two- and
three-way data analysis. Paper presented at the Seventh Conference of
the
**International Federation of Classification Societies**, Namur, Belgium, July, 2000. (invited) - Hong, S., & Harshman, R. A. Parafac2
fitting of covariance matrices from pooled and nested datasets with an
adjustment for factor mean differences. Paper presented at the Seventh
Conference of the
**International Federation of Classification Societies**, Namur, Belgium, July, 2000. - Harshman, R. A. A generalization of matrix notation and algebra to n-way arrays. Paper presented at
**TRICAP 2000**, a conference on three-way methods in chemistry and psychology, Faaborg, Denmark, July, 2000. (invited) - Hong, S., & Harshman, R. A. Shifted factor analysis: Algorithms and applications. Paper presented at
**TRICAP 2000**, a conference on three-way methods in chemistry and psychology, Faaborg, Denmark, July, 2000. - Swartzman, L. C., Harshman, R. A., Burkell,
J., & Power, T. E. What is the basis upon which people distinguish
between "Traditional" and "Natural" treatment approaches? Poster
presented at the Twenty-first Annual Sessions of the
**Society of Behavioral Medicine**, Nashville, Tennessee, April, 2000. - Johnson,
A. M., MacDonald, P. L., Harshman, R. A., Vernon, P. A., &
Paunonen, S. V. Biological factor analysis: A three-way (PARAFAC)
analysis of twin data. Poster presented at the Annual Meeting of the
**Behavior Genetics Society**, Toronto, July, 1997. - Bro, R., & Harshman, R. A. Constraints in multiway analysis. Paper presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May, 1997. - Harshman, R. A. Constrained PARAFAC and
PARATUCK3 models for genetic study of multivariate data from fraternal
vs identical twins. Poster presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May, 1997. - Harshman, R. A. Increasing the flexibility of three-way analysis: "Model morphing" and "elastic fitting." Paper presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May, 1997. (invited) - Harshman, R. A., & Lundy, M. E. An
"extended PARAFAC model" incorporating singly-subscripted constants.
Poster presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May, 1997.View .pdf file (200KB, 8 pp) - Hong, S., & Harshman, R. A. Shifted-factor analysis:
Decomposing mixtures of curves into underlying components that not only
change in size but also shift along the frequency or time axis. Paper
presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May 1997. - Hopke, P. K., Paatero, P., Jia, H., Ross, R. T., &
Harshman, R. A. Three-way factor analysis: Examination and comparison
of alternative computational methods. Paper presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May, 1997. - Paatero, P., Harshman, R. A., & Lundy, M. E. Synthetic degenerate PARAFAC models:
Construction, properties and connections with centering. Poster presented at
**TRICAP '97**, a conference on three-way methods in chemistry and psychology, Lake Chelan, Washington, May, 1997. - Harshman, R. A., & Chino, N. A refactoring method for
fitting the Hermitian factor analysis model that allows self-similarity
estimation and missing values. Paper presented at the Twenty-fourth
Annual Meeting of the
**Behaviormetric Society of Japan**, Chiba, Japan, September, 1996. - Do, T., Harshman, R. A., McIntyre, N. S., & Lundy, M. E.
New application of parallel factor analysis in the study of oxidation
of aluminum by using X-ray photoelectron spectroscopy. Poster presented
at the Eighth
**Canadian Materials Science**Conference, London, Ontario, June, 1996.

- Harshman, R. A. Generalizations, extensions
and structured special-purpose modifications of PARAFAC with possible
applications in chemistry. Paper presented at the First Conference on
**Three-Way Analysis Methods in Chemistry (TRIC)**: A Meeting of Psychometrics and Chemometrics, Epe, The Netherlands, August, 1993. - Harshman, R. A., & Lundy, M. E.
Three-way DEDICOM: Analyzing multiple matrices of asymmetric
relationships. Paper presented at the Annual Meeting of the
**North American Psychometric Society**, Columbus, Ohio, July, 1992. - Harshman, R. A., & Lundy, M. E.
"Oblique and still unique?" Applying parallel profiles in more general
models for factor analysis and multidimensional scaling. Paper
presented at the Joint Annual Meeting of the
**Classification and Psychometric Societies**, New Brunswick, New Jersey, June, 1991. - Harshman, R. A., Lundy, M. E., &
Kruskal, J. B. Comparison of trilinear and quadrilinear methods:
Strengths, weaknesses, and degeneracies. Paper presented at
**"Multiway '88"**, an international meeting on the analysis of multiway data matrices, Rome, March, 1988. (invited) - Kruskal, J. B., Harshman, R. A., &
Lundy, M. E. Some relationships among Tucker three-mode factor analysis
(3-MFA), PARAFAC-CANDECOMP, and CANDELINC. Paper presented at
**"Multiway '88"**, an international meeting on the analysis of multiway data matrices, Rome, March, 1988. - Harshman, R. A., & Kiers, H. Algorithms
for DEDICOM analysis of asymmetric data. Paper presented at the
European Meeting of the
**Psychometric Society**, Enschede, The Netherlands, June, 1987. - Harshman, R. A., & Lundy, M. E. "New"
methods of exploratory factor analysis use three-way data to solve
rotation problem. Poster presented at the Third Meeting of the
**International Society for the Study of Individual Differences**, Toronto, Ontario, June, 1987. - Dawson, M. R., & Harshman, R. A.
Multidimensional analysis of asymmetries in stimulus confusions. Paper
presented at the Annual Meeting of the
**Psychometric Society**, Toronto, June, 1986. - Harshman, R. A., Lundy, M. E., &
Kruskal, J. B. Centering of three-way and two-way data: Theory and
diagnostics. Paper presented at the Annual Meeting of the
**Psychometric Society**, Toronto, June, 1986. - Kinnucan, M. T., & Harshman, R. A.
Symmetric and skew-symmetric components of alphabetic confusion
matrices. Paper presented at the Annual Meeting of the
**Classification Society of North America**, Columbus, Ohio, June, 1986. - Harshman, R. A., and Lundy, M. E. Multidimensional analysis of preference structures. Presented at the
**Bell Communications Research Telecommunications Demand Modeling Conference**, New Orleans, October, 1985. - Kruskal, J. B., Harshman, R. A., &
Lundy, M. E. Several mathematical relationships between
PARAFAC-CANDECOMP and three-mode factor analysis. Presented at the
Annual Meeting of the
**Classification Society of North America**, St. John's, Newfoundland, July, 1985. - Lundy, M. E., Harshman, R. A., &
Kruskal, J. B. A two-stage procedure incorporating good features of
both trilinear and quadrilinear methods. Presented at the Annual
Meeting of the
**Classification Society of North America**, St. John's, Newfoundland, July, 1985. - Harshman, R. A. (1978). Models for
analysis of asymmetrical relationships among N objects or stimuli.
Paper presented at the First Joint Meeting of the
**Psychometric Society**and**The Society for Mathematical Psychology**, Hamilton, Ontario, August.View .pdf file (1340KB, 25 pp)

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