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Finding the limit of diverging components in three-way Candecomp/Parafac—A demonstration of its practical merits

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  • Stegeman, Alwin

Abstract

Three-way Candecomp/Parafac (CP) is a three-way generalization of principal component analysis (PCA) for matrices. Contrary to PCA, a CP decomposition is rotationally unique under mild conditions. However, a CP analysis may be hampered by the non-existence of a best-fitting CP decomposition with R≥2 components. In this case, fitting CP to a three-way data array results in diverging CP components. Recently, it has been shown that this can be solved by fitting a decomposition with several interaction terms, using initial values obtained from the diverging CP decomposition. The new decomposition is called CPlimit, since it is the limit of the diverging CP decomposition. The practical merits of this procedure are demonstrated for a well-known three-way dataset of TV-ratings. CPlimit finds main components with the same interpretation as Tucker models or when imposing orthogonality in CP. However, CPlimit has higher joint fit of the main components than Tucker models, contains only one small interaction term, and does not impose the unnatural constraint of orthogonality. The uniqueness properties of the CPlimit decomposition are discussed in detail.

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  • Stegeman, Alwin, 2014. "Finding the limit of diverging components in three-way Candecomp/Parafac—A demonstration of its practical merits," Computational Statistics & Data Analysis, Elsevier, vol. 75(C), pages 203-216.
    • Handle: RePEc:eee:csdana:v:75:y:2014:i:c:p:203-216
    DOI: 10.1016/j.csda.2014.02.010
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    References listed on IDEAS

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    1. Wim Krijnen & Theo Dijkstra & Alwin Stegeman, 2008. "On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model," Psychometrika, Springer;The Psychometric Society, vol. 73(3), pages 431-439, September.
    2. J. Carroll & Jih-Jie Chang, 1970. "Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition," Psychometrika, Springer;The Psychometric Society, vol. 35(3), pages 283-319, September.
    3. Alwin Stegeman, 2007. "Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank," Psychometrika, Springer;The Psychometric Society, vol. 72(4), pages 601-619, December.
    4. Tomasi, Giorgio & Bro, Rasmus, 2006. "A comparison of algorithms for fitting the PARAFAC model," Computational Statistics & Data Analysis, Elsevier, vol. 50(7), pages 1700-1734, April.
    5. Kiers, Henk A. L., 1998. "Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure," Computational Statistics & Data Analysis, Elsevier, vol. 28(3), pages 307-324, September.
    6. Henk Kiers, 2006. "Properties of and Algorithms for Fitting Three-Way Component Models with Offset Terms," Psychometrika, Springer;The Psychometric Society, vol. 71(2), pages 231-256, June.
    7. Pieter Kroonenberg & Jan Leeuw, 1980. "Principal component analysis of three-mode data by means of alternating least squares algorithms," Psychometrika, Springer;The Psychometric Society, vol. 45(1), pages 69-97, March.
    8. Ledyard Tucker, 1966. "Some mathematical notes on three-mode factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 31(3), pages 279-311, September.
    9. Harshman, Richard A. & Lundy, Margaret E., 1994. "PARAFAC: Parallel factor analysis," Computational Statistics & Data Analysis, Elsevier, vol. 18(1), pages 39-72, August.
    10. Siciliano, Roberta & Mooijaart, Ab, 1997. "Three-factor association models for three-way contingency tables," Computational Statistics & Data Analysis, Elsevier, vol. 24(3), pages 337-356, May.
    11. Alwin Stegeman, 2006. "Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher," Psychometrika, Springer;The Psychometric Society, vol. 71(3), pages 483-501, September.
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    1. Alwin Stegeman, 2018. "Simultaneous Component Analysis by Means of Tucker3," Psychometrika, Springer;The Psychometric Society, vol. 83(1), pages 21-47, March.

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