## Uniqueness of real-valued hierarchical classes models.(English)Zbl 1204.91114

Summary: Two novel uniqueness theorems are derived for the family of hierarchical classes (HICLAS) models, a family of structural decomposition models for $$N$$-way $$N$$-mode data that imply simultaneous hierarchically organized classifications of all modes involved in the data. The theorems generalize earlier results on binary HICLAS models to the integer- and real-valued cases. In addition, they allow for a shorter and insightful proof of a result on Boolean matrix invertibility that goes back to earlier work of R. D. Luce [Proc. Am. Math. Soc. 3, 382–388 (1952; Zbl 0048.02302)] and D. E. Rutherford [Proc. Glasg. Math. Assoc. 6, 49–53 (1963; Zbl 0114.01701)].

### MSC:

 91C15 One- and multidimensional scaling in the social and behavioral sciences 62J15 Paired and multiple comparisons; multiple testing 15B34 Boolean and Hadamard matrices 62P12 Applications of statistics to environmental and related topics

### Keywords:

hierarchical classes; uniqueness; RV-HICLAS

### Citations:

Zbl 0048.02302; Zbl 0114.01701

Tucker3-HICLAS
Full Text:

### References:

 [1] Carroll, J.D.; Arabie, P., Multidimensional scaling, Annual review of psychology, 31, 607-649, (1980) [2] Carroll, J.D.; Chang, J.J., Analysis of individual differences in multidimensional scaling via an $$N$$-way generalization of “eckart – young” decomposition, Psychometrika, 35, 283-319, (1970) · Zbl 0202.19101 [3] Ceulemans, E.; Van Mechelen, I., Uniqueness of $$N$$-way $$N$$-mode hierarchical classes models, Journal of mathematical psychology, 47, 259-264, (2003) · Zbl 1052.91075 [4] Ceulemans, E.; Van Mechelen, I.; Leenen, I., Tucker3 hierarchical classes analysis, Psychometrika, 68, 413-433, (2003) · Zbl 1306.62393 [5] De Boeck, P.; Rosenberg, S., Hierarchical classes: model and data analysis, Psychometrika, 53, 361-381, (1988) · Zbl 0718.62001 [6] Harshman, R.A. (1970). Foundations of the parafac procedure: Models and conditions for an explanatory multi-modal factor analysis. {\scucla} Working papers in phonetics, vol. 16, pp. 1-84. [7] Kiers, H.A.L., Hierarchical relations among three-way methods, Psychometrika, 56, 449-470, (1991) · Zbl 0760.62059 [8] Kiers, H.A.L., Towards a standardized notation and terminology in multiway analysis, Journal of chemometrics, 14, 105-122, (2000) [9] Kiers, H.A.L.; Van Mechelen, I., Three-way component analysis: principles and illustrative application, Psychological methods, 6, 84-110, (2001) [10] Kim, K.H., Boolean matrix theory, (1982), Marcel Dekker New York [11] Kroonenberg, P.M.; De Leeuw, J., Principal component analysis of three-mode data by means of alternating least squares algorithms, Psychometrika, 45, 69-97, (1980) · Zbl 0431.62035 [12] Leenen, I.; Van Mechelen, I.; De Boeck, P.; Rosenberg, S., {\scindclas}: A three-way hierarchical classes model, Psychometrika, 64, 9-24, (1999) · Zbl 1365.62456 [13] Luce, R.D., A note on Boolean matrix theory, Proceedings of the American mathematical society, 3, 382-388, (1952) · Zbl 0048.02302 [14] Rutherford, D.E., Inverse of Boolean matrices, Proceedings of the Glasgow mathematical association, 6, 49-53, (1963) · Zbl 0114.01701 [15] Schepers, J., & Van Mechelen, I. (2008). The real-valued model of hierarchical classes (submitted for publication). · Zbl 1360.62351 [16] Tucker, L.R., Some mathematical notes on three-mode factor analysis, Psychometrika, 31, 279-311, (1966) [17] Van Mechelen, I.; Bock, H.-H.; De Boeck, P., Two-mode clustering methods: A structured overview, Statistis in medicine, 13, 363-394, (2004) · Zbl 1053.62078 [18] Van Mechelen, I.; De Boeck, P.; Rosenberg, S., The conjunctive model of hierarchical classes, Psychometrika, 60, 505-521, (1995) · Zbl 0864.92021 [19] Van Mechelen, I.; Lombardi, L.; Ceulemans, E., Hierarchical classes modeling of rating data, Psychometrika, 72, 475-488, (2007) · Zbl 1291.62248
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.