Uniqueness of \(N\)-way \(N\)-mode hierarchical classes models. (English) Zbl 1052.91075
Summary: This paper presents two uniqueness theorems for the family of hierarchical classes models, a collection of order preserving Boolean decomposition models for binary \(N\)-way \(N\)-mode data. The theorems are compared with uniqueness results for the closely related family of \(N\)-way \(N\)-mode principal component models. It is concluded that the two-way two-mode PCA and \(N\)-way \(N\)-mode Tucker\(N\) models suffer more from a lack of identifiability than their hierarchical classes analogues, whereas the uniqueness conditions for \(N\)-way \(N\)-mode PARAFAC/CANDECOMP models are less restrictive than the ones derived for their \(N\)-way \(N\)-mode hierarchical classes counterparts.
MSC:
| 91C15 | One- and multidimensional scaling in the social and behavioral sciences |
References:
| [1] | Bro, R.; De Jong, S., A fast non-negativity-constrained least squares algorithm, Journal of Chemometrics, 11, 393-401 (1997) |
| [2] | Bro, R.; Sidiropoulos, N. D., Least squares algorithms under unimodality and non-negativity constraints, Journal of Chemometrics, 12, 223-247 (1998) |
| [3] | 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 |
| [4] | Carroll, J. D., & Pruzansky, S. (1984). The Candecomp-Candelinc family of models and methods for multidimensional data analysis. In H. G. Law, J. C. W. Snyder, J. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis; Carroll, J. D., & Pruzansky, S. (1984). The Candecomp-Candelinc family of models and methods for multidimensional data analysis. In H. G. Law, J. C. W. Snyder, J. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis |
| [5] | Ceulemans, E., Van Mechelen, I., & Leenen, I (in press). Tucker3 hierarchical classes analysis. Psychometrika; Ceulemans, E., Van Mechelen, I., & Leenen, I (in press). Tucker3 hierarchical classes analysis. Psychometrika · Zbl 1306.62393 |
| [6] | De Boeck, P.; Rosenberg, S., Hierarchical classesModel and data analysis, Psychometrika, 53, 361-381 (1988) · Zbl 0718.62001 |
| [7] | Falmagne, J.-C; Koppen, M.; Vilano, M.; Doignon, J.-P; Johannesen, L., Introduction to knowledge spacesHow to build, test and search them, Psychological Review, 97, 201-224 (1990) |
| [8] | Gara, M.; Rosenberg, S., A set-theoretical model of person perception, Behavioral research, 25, 275-293 (1990) |
| [9] | Harshman, R. A. (1970). Foundations of the parafacuclaWorking Papers in Phonetics; Harshman, R. A. (1970). Foundations of the parafacuclaWorking Papers in Phonetics |
| [10] | Harshman, R. A., & Lundy, M. E. (1984). The Parafac model for three-way factor analysis and multidimensional scaling. In H. G. Law, J. C. W. Snyder, J. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis; Harshman, R. A., & Lundy, M. E. (1984). The Parafac model for three-way factor analysis and multidimensional scaling. In H. G. Law, J. C. W. Snyder, J. Hattie, & R. P. McDonald (Eds.), Research methods for multimode data analysis |
| [11] | Kiers, H. A.L, Towards a standardized notation and terminology in multiway analysis, Journal of Chemometrics, 14, 105-122 (2000) |
| [12] | Kiers, H. A.L; Smilde, A. K., Constrained three-mode factor analysis as a tool for parameter estimation with second-order instrumental data, Journal of Chemometrics, 12, 125-147 (1998) |
| [13] | Kiers, H. A.L; Ten Berge, J. M.F; Rocci, R., Uniqueness of three-mode factor models with sparse coresThe \(3^∗3^∗3\) case, Psychometrika, 62, 349-374 (1997) · Zbl 0890.62046 |
| [14] | Kim, K. H., Boolean matrix theory (1982), Marcel Dekker: Marcel Dekker New York · Zbl 0495.15003 |
| [15] | Kruskal, J. B., Three-way arraysRank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics, Linear Algebra and its Applications, 18, 95-138 (1977) · Zbl 0364.15021 |
| [16] | Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and \(n\)Multiway data analysis; Kruskal, J. B. (1989). Rank, decomposition, and uniqueness for 3-way and \(n\)Multiway data analysis |
| [17] | Leenen, I.; Van Mechelen, I.; De Boeck, P.; Rosenberg, S., indclasA three-way hierarchical classes model, Psychometrika, 64, 9-24 (1999) · Zbl 1365.62456 |
| [18] | Luce, R. D., A note on Boolean matrix theory, Proceedings of the American Mathematical Society, 3, 382-388 (1952) · Zbl 0048.02302 |
| [19] | Rutherford, D. E., Inverse of Boolean matrices, Proceedings of the Glasgow Mathematical Association, 6, 49-53 (1963) · Zbl 0114.01701 |
| [20] | Sidiropoulos, N. D.; Bro, R., On the uniqueness of multilinear decomposition of \(n\)-way arrays, Journal of Chemometrics, 14, 229-239 (2000) |
| [21] | Tucker, L. R., Some mathematical notes on three-mode factor analysis, Psychometrika, 31, 279-311 (1966) |
| [22] | Van Mechelen, I.; De Boeck, P., Implicit taxonomy in psychiatric diagnosisA case study, Journal of Social and Clinical Psychology, 8, 276-287 (1989) |
| [23] | Van Mechelen, I.; De Boeck, P.; Rosenberg, S., The conjunctive model of hierarchical classes, Psychometrika, 60, 505-521 (1995) · Zbl 0864.92021 |
| [24] | Van Mechelen, I.; Van Damme, G., A latent criteria model for choice data, Acta Psychologica, 87, 85-94 (1994) |
| [25] | Vansteelandt, K.; Van Mechelen, I., Individual differences in situation-behavior profilesA triple typology model, Journal of Personality and Social Psychology, 75, 751-765 (1998) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
