This paper presents a new measure of the central point of population using two-dimensional normal distribution theory. When a previously presented measure, for example, the center of population (equation (1)) is calculated for one city, its value does not necessarily indicate a singular point for that city. This is because previous measures depend on how an observer delimits the city area. The center of concentric contour circles of population density, however, should indicate the singular point for the city, if its population forms a normal distribution. The position of this center can be estimated from the two-dimensional normal distribution type of model (equation (3)), since a pair of variables (xc, yc) in the model are identified as the coordinates of the center. Thus, this paper defines a new measure as a pair of the least square estimates (xc, yc) and calls this measure the singular point of urban population. The estimates xc and yc are part of (dc, a, xc, yc) which minimizes the error sum of squares S in equation (5). The variable di in (5) is an observed value of the population density at place (xi, yi).
Although (xc, yc) in (3) is equivalent to the mean value in a two-dimensional normal distribution, Figure 1 shows that (xc, yc) in (5) is different from the mean value in the sample distribution. Accordingly, the singular point of urban population is not a concept included in “ordinary” statistics, but a concept peculiar to spatial science.
The iterative method is necessary for estimating the point, since (3) is a nonlinear regression equation. This method, however, is more complicated than the algebraic method. Thus, this paper introduces the following two methods to obtain an approximate algebraic solution. One is to derive equation (7) from (3) using logarithmic linearization, and then to obtain an approximate value from (7) using linear regression analysis. This value is given by equations (12) and (13), where f=logd; z=x2+y2; Sfx=Σ(fi-f)(xi-x); and Cij is a cofactor of the square matrix in equation (8). The other is to obtain a more approximate value from (7) using weighted regression analysis. This value is given by equations (15) and (16), where Tdfx=ΣdiΣdifixi-ΣdifiΣdixi; and C'ij is a cofactor of the square matrix in equation (14). In this paper, the above three kinds of values calculated by nonlinear, linear, and weighted regression analysis are called the nonlinear solution, the linear solution, and the weighted solution respectively.
These solutions and the center of population are applied to the following areas: two hypothetical cities where population forms a concentric distribution (city A) or an elliptic distribution (city B); and three actual cities in 1970, 1975, and 1980 (Utsunomiya, Koriyama, and Yamagata). Figures 2 and 3 show the limits of these cities. Hypothetical cities A and B are delimited in two and four ways respectively, so that the city limits of A0, A1, B0, B1, B2, and B3 are established; the limits of the actual cities are established within each circle centered at the central station. In the hypothetical cities, the true value of the singular point of urban population is determined at the origin (0.0, 0.0); nevertheless in the actual cities
