SPATIAL ANALYSIS OF DIFFUSION PROCESS BY MULTI-DIMENSIONAL SCALING

The purpose of this paper is to consider diffusion problems in time-space. It also aims at illustrating the usefulness of spatial analysis in terms of metric transformation. Previous works suggest that another type of distance such as time-distance or social distance is more relevant to analyze diffusion process rather than physical distance, but no positive work seems to exist. So, in this paper, inter-urban diffusion of the 1957 epidemic of Asian influenza in Nagoya and its environs is studied by recovering time-space. This study is concerned with 21 cities with more than fifty thousands population as of 1960. The epidemic hit 20 cities except Nishio, among which the first outbreaks were reported in Nagoya and Kasugai on May 24 (Fig. 2).<br> First of all, we must present time-space before examining the diffusion process. Timespace and physical space, by definition, are those recovered by application of M-D-SCAL, an algorithm of non-metric multi-dimensional scaling, to matrices of railway and/or bus travel times and straight distances between every pair of 21 cities respectively. In recovering each space, the following assumptions were made<br> (1) Two-dimensional solution is sought.<br> (2) Interpoint distances are measured by Euclidean distance.<br> (3) The initial value of step-size is set at 0.2.<br> (4) Seven starting coordinates are used, that is, five random coordinates, the coordi nates based on the metric method of Torgerson and the ordinary map coordinates.<br> (5) In order to get the configuration from which no further improvement is possible, 100 iterations are equally made for each case.<br> (6) The minimum-stress configuration derived from seven starting coordinates is re garded as fitting the data best, and used for the later analysis.<br> Table 2 shows both stresses of physical space and time-space derived from seven starting coordinates. For physical space, the starting coordinates based on the metric method of Torgerson yielded the minimum stress, nearly 0. 0. Its corresponding two-dimensional configuration is shown in Fig. 3. The first dimension could be interpreted roughly as representing SE-NW direction and the second dimension as representing NE-SW direction. Fig. 4 represents the two-dimensional configuration of time-space with the minimum stress (14. 907%), whose starting coordinates are the second random one. Locations of cities by quadrants characterize the derived two-dimensional time-space: Gifu, Ogaki, Ichinomiya and so on in the western Owari and the Seino districts are located in the top right-hand quadrant; cities in the eastern Owari and the Tono districts as well as Nagoya are located in the top left-hand quadrant; Handa and Tokoname in the Owari-Chita district are located in the bottom left-hand quadrant; Toyohashi, Okazaki and so on in the Mikawa district are located in the bottom right-hand quadrant. Compared Fig. 4 with Fig. 2 and Fig. 3, however, not only locations of cities but also the relative locations of districts don't coincide with those in the ordinary map. This is typically indicated by the fact that the locational relationship between the western Owari and the Seino districts, and the eastern Owari and the Tono districts is reversed. Such difference between time-space and ordinary map would be ascribed to the imperfectness of inter-district railways systems in the study area. For example, there exists no railway directly connecting the Tono district and the Mikawa district.