Abstract
Multidimensional scaling aims at reconstructing dissimilarities between pairs of objects by distances in a low dimensional space. However, in some cases the dissimilarity itself is not known, but the range, or a histogram of the dissimilarities is given. This type of data fall in the wider class of symbolic data (see Bock and Diday (2000)). We model three-way two-mode data consisting of an interval of dissimilarities for each object pair from each of K sources by a set of intervals of the distances defined as the minimum and maximum distance between two sets of embedded rectangles representing the objects. In this paper, we provide a new algorithm called 3WaySym-Scal using iterative majorization, that is based on an algorithm, I-Scal developed for the two-way case where the dissimilarities are given by a range of values ie an interval (see Groenen et al. (2006)). The advantage of iterative majorization is that each iteration is guaranteed to improve the solution until no improvement is possible. We present the results on an empirical data set on synthetic musical tones.
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References
BOCK, H.-H. and DIDAY, E. (2000): Analysis of Symbolic Data Springer, Berlin.
BORG, I. and GROENEN, P.J.F. (2005): Modern Multidimensional Scaling: Theory and Applications, Second Edition Springer, New York.
CARROLL, J.D. and CHANG, J.J. (1972): Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart-Young decomposition. Psyhometika, 35, 283–319.
DE LEEUW, J. and HEISER, W.J. (1980): Multidimensional scaling with restrictions on the configuration. In: P. R. Krishnaiah (Ed.), Multivariate analysis (Vol. V) (pp. 501–522). Amsterdam, The Netherlands: North-Holland.
DENŒUX, T. and MASSON, M. (2000): Multidimensional scaling of interval-valued dissimilarity data. Pattern Recognition Letters, 21, 83–92.
GROENEN, P.J.F., WINSBERG, S., RODRIGUEZ, O. and DIDAY, E. (2006): I-Scal: Multidimensional scaling of interval dissimilarities. Computational Statistics and Data Analysis, 51, 360–378.
GROENEN, P.J.F., and WINSBERG, S. (2006): Multidimensional scaling of histogram dissimilarities. In: V. Batagelj, H.-H. Bock, A. Ferligoj, and A. Ziberna (Eds.), Proceedings of the ICFS Llubjana, Slovenia (pp. 161–170). Springer, Berlin.
KRUSKAL, J.B. (1964): Multidimensional scaling by optimizing goodness of fit to a nonmetric hypothesis. Psychometrika, 29, 1–27.
MASSON, M. and DENOEUX, T. (2002): Multidimensional scaling of fuzzy dissimilarity data; Fuzzy Sets and Systems, 128, 339–352.
MCADAMS, S. and WINSBERG, S. [1999]: Multidimensional scaling of musical timbre constrained by physical parameters. The Journal of the Acoustical Society of America, 105, 1273.
MCADAMS, S., WINSBERG, S., DONNADIEU, S., DESOETE, G., and KRIMPHOFF, J. (1995) Perceptual scaling of synthesized musical timbres: Common dimensions, specificities, and latent subject classes. Psychological Research, 58, 177–192.
PLOMP, R. (1970): Timbre as a multidimensional attribute of complex tones. In: R. Plomp & G.F. Smoorenburg (Eds.) Frequency analysis and periodicity detection in hearing (pp 397–414. Leiden: Sijthoff
WINSBERG, S. and CARROLL, J.D. (1989): A quasi-nonmetric method for multidimensional scaling via an extended Euclidean model. Psychomtrika, 54, 217–229.
WINSBERG, S. and DESOETE, G. (1993): A latent class approach to fitting the weighted Euclidean model, CLASCAL. Psychometika, 58, 31–330.
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Groenen, P.J.F., Winsberg, S. (2007). 3WaySym-Scal: Three-Way Symbolic Multidimensional Scaling. In: Brito, P., Cucumel, G., Bertrand, P., de Carvalho, F. (eds) Selected Contributions in Data Analysis and Classification. Studies in Classification, Data Analysis, and Knowledge Organization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73560-1_6
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DOI: https://doi.org/10.1007/978-3-540-73560-1_6
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