Abstract
A new procedure is discussed which represents the asymmetric relationships between N objects. These relationships must be defined at the interval levels of measurement in Steven’s terminology. This method not only reveals the clustering of the objects but also enables us to give information about both the magnitude and the orientation of the “skewness” between objects in “a” configuration. The procedure optimizes the fit of the model directly to the data by an alternating least squares procedure. It is found to be robust, as the Hessian Matrix of the loss function is positive definite at least in two dimensional case, except for a special case. The method is illustrated with an artificial data and three empirical data.
Similar content being viewed by others
References
Bard, Y. (1974). Nonlinear parameter estimation. Academic Press.
Box, M.J. (1966). A comparison of several current optimization methods, and the use of transformations in constrained problems. The Computer Journal, 9, 67–77.
Chino, N. (1977). N-ko no taisho-kan no hitaisho na kankei o zusikika suru tame no ichi-giho. Proceedings of the 5-th annual conference of the Behaviormetric Society of Japan at Okayama University, 146–149.
Coombs, C.H. (1964). A theory of data. Wiley.
Coombs, C.H., Dawes, R.M. & Tversky, A. (1970). Mathematical Psychology. Prentice Hall.
Guttman, L. (1968). A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika, 33, 469–506.
Hayashi, C. (1952). On the Prediction of Phenomena from Qualitative Data and the Quantification of Qualitative Data from the Mathematico-Statistical Point of View. Annals of the Institute of Statistical Mathematics, 3, No. 2, 69–98.
Hayashi, C. (1974). Minimum dimension analysis. Behaviormetrika, 1, 1–24.
Hayashi, C. (1977). MDS ni okeru hitaishosei no mondai. Proceedings of the 45-th annual conference of Japan Statistical Society at Fukuoka University, 41.
Lingoes, J.C. (1973). The Guttman-Lingoes nonmetric program series. Ann Arbor: Mathesis Press.
Takane, Y., Young, F.W., & de Leeuw, J. (1977). Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7–67.
Young, F.W. (1975). An asymmetric Euclidian model for multi-process asymmetric data. Paper distributed at U.S.-Japan Seminar on MDS, San Diego.
Author information
Authors and Affiliations
About this article
Cite this article
Chino, N. A Graphical Technique for Representing the Asymmetric Relationships Between N Objects. Behaviormetrika 5, 23–40 (1978). https://doi.org/10.2333/bhmk.5.23
Received:
Published:
Issue Date:
DOI: https://doi.org/10.2333/bhmk.5.23