**ERIC Number:**EJ785466

**Record Type:**Journal

**Publication Date:**2007-Dec

**Pages:**19

**Abstractor:**Author

**Reference Count:**0

**ISBN:**N/A

**ISSN:**ISSN-0033-3123

Degeneracy in Candecomp/Parafac and Indscal Explained for Several Three-Sliced Arrays with a Two-Valued Typical Rank

Stegeman, Alwin

Psychometrika, v72 n4 p601-619 Dec 2007

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and some elements of the rank-1 arrays become arbitrarily large. We consider the real-valued CP decomposition of all known three-sliced arrays, i.e., of size p[times]q[times]3, with a two-valued typical rank. These are the 5[times]3[times]3 and 8[times]4[times]3 arrays, and the 3[times]3[times]4 and 3[times]3[times]5 arrays with symmetric 3[times]3 slices. In the latter two cases, CP is equivalent to the Indscal model. For a typical rank of {m,m+1}, we consider the CP decomposition with R=m of an array of rank m+1. We show that (in most cases) the CP objective function does not have a minimum but an infimum. Moreover, any sequence of feasible CP solutions in which the objective value approaches the infimum will become degenerate. We use the tools developed in Stegeman (2006), who considers p[times]p[times]2 arrays, and present a framework of analysis which is of use to the future study of CP degeneracy related to a two-valued typical rank. Moreover, our examples show that CP uniqueness is not necessary for degenerate solutions to occur.

Descriptors: Research Methodology, Data Analysis, Models, Psychological Studies, Psychometrics, Evaluation Methods

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A