multiway-package: Component Models for Multi-Way Data

Description Details Author(s) References Examples


Fits multi-way component models via alternating least squares algorithms with optional constraints (orthogonality and non-negativity). Fit models include Individual Differences Scaling, Parallel Factor Analysis (1 and 2), Simultaneous Component Analysis, and Tucker Factor Analysis.


indscal fits the Individual Differences Scaling model. parafac fits the 3-way and 4-way Parallel Factor Analysis-1 model. parafac2 fits the 3-way and 4-way Parallel Factor Analysis-2 model. sca fits the four different Simultaneous Component Analysis models. tucker fits the 3-way and 4-way Tucker Factor Analysis model.


Nathaniel E. Helwig <>

Maintainer: Nathaniel E. Helwig <>


Bro, R., & De Jong, S. (1997). A fast non-negativity-constrained least squares algorithm. Journal of Chemometrics, 11, 393-401.

Bro, R., & Kiers, H.A.L. (2003). A new efficient method for determining the number of components in PARAFAC models. Journal of Chemometrics, 17, 274-286.

Carroll, J. D., & Chang, J-J. (1970). Analysis of individual differences in multidimensional scaling via an n-way generalization of "Eckart-Young" decmoposition. Psychometrika, 35, 283-319.

Harshman, R. A. (1970). Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multimodal factor analysis. UCLA Working Papers in Phonetics, 16, 1-84.

Harshman, R. A. (1972). PARAFAC2: Mathematical and technical notes. UCLA Working Papers in Phonetics, 22, 30-44.

Harshman, R. A., & Lundy, M. E. (1994). PARAFAC: Parallel factor analysis. Computational Statistics and Data Analysis, 18, 39-72.

Helwig, N. E. (2013). The special sign indeterminacy of the direct-fitting Parafac2 model: Some implications, cautions, and recommendations, for Simultaneous Component Analysis. Psychometrika, 78, 725-739.

Kiers, H. A. L., ten Berge, J. M. F., & Bro, R. (1999). PARAFAC2-part I: A direct-fitting algorithm for the PARAFAC2 model. Journal of Chemometrics, 13, 275-294.

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.

Moore, E.H. (1920). On the reciprocal of the general algebraic matrix. Bulletin of the American Mathematical Society 26, 394-395.

Penrose, R. (1950). A generalized inverse for matrices. Mathematical Proceedings of the Cambridge Philosophical Society 51, 406-413.

Timmerman, M. E., & Kiers, H. A. L. (2003). Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences. Psychometrika, 68, 105-121.

Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279-311.


# See examples for indscal, parafac, parafac2, sca, and tucker

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