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Top1. Introduction
Asymmetric dissimilarity data is data such that dissimilarity from subject i to subject j and dissimilarity from j to i are not the same necessarily. If the value of the dissimilarity from i to j is larger than that from j to i, we consider that the dissimilarity from i to j is relatively far comparison with from j to i. There are various situations such that asymmetric dissimilarity data is observed. For example, brand switching data is observed as an asymmetric similarity data in marketing domains. In the brand switching data, the element from brand i to j indicates frequency of buying brand j after buying brand i for customers. To reveal the competitive relations among brands, Asymmetric Multidimensional Scaling (AMDS) is applied to the brand switching data (e.g. Okada and Tsurumi, 2012), where AMDS is visualization method for the asymmetric relations for subjects. In usually, brand switching data is generated from POS data. Therefore, we can usually get the multivariate data as external information for the same subjects along with the brand switching data from the POS data. However, in such the situation, ordinal AMDS provide only the interpretation of asymmetric relations, not the relation between the asymmetries and the external information. If the estimated axes in AMDS can be interpreted as those in Principal Component Analysis (PCA), the visualization result is very useful. Therefore, we propose new AMDS with external information.
Several models of AMDS have been proposed (Borg and Groenen, 2005; Chino, 1978, 1990, 2012; Chino and Shiraiwa, 1993; Harshman, 1978; Harshman et al, 1982; Kiers and Takane 1994; Krumhansl, 1978; Loisel and Takane (2011); Okada and Imaizumi, 1984, 1987, 1997; Rocci and Bove, 2002; Saito and Yadohisa, 2005; Yadohisa and Niki, 1999). Chino (2012) classify these models into groups. In Unfolding type models among these models (Constantine and Gower, 1978; De Leeuw and Heiser, 1982; Gower, 1977; Zietlman and Heiser, 1993, 1996), each subject on the estimated low dimensions can be described by two different coordinates to represent the asymmetric relation. There are two kinds of Unfolding type models such as Unfolding and slide-vector model. The slide-vector model is considered as a parsimonious model of Unfolding model. The advantage of these methods is that it is easy to interpret the asymmetric relations in Euclidean space. In this paper, we propose new AMDS with external information based on path structures. Given asymmetric dissimilarity data, the multivariate data for the same subjects and path structure, the proposed method provide the low dimensions such that asymmetric relations and each axis can be interpreted like PCA or Generalized Structured Component Analysis (GSCA) (Hwang and Takane, 2004). For the image of the proposed method, see Figure 1. Therefore, the proposed method can reflect the assumption of researchers through the path structure. This is the advantage of the proposed method.
Figure 1. Image of AMDS with external information based on path structure