Statistical distribution of building lot frontage: application for Tokyo downtown districts

Abstract

The frontage of a building lot is the determinant factor of the residential environment. The statistical distribution of building lot frontages shows how the perimeters of urban blocks are shared by building lots for a given density of buildings and roads. For practitioners in urban planning, this is indispensable to identify potential districts which comprise a high percentage of building lots with narrow frontage after subdivision and to reconsider the appropriate criteria for the density of buildings and roads as residential environment indices. In the literature, however, the statistical distribution of building lot frontages and the density of buildings and roads has not been fully researched. In this paper, based on the empirical study in the downtown districts of Tokyo, it is found that (1) a log-normal distribution fits the observed distribution of building lot frontages better than a gamma distribution, which is the model of the size distribution of Poisson Voronoi cells on closed curves; (2) the statistical distribution of building lot frontages statistically follows a log-normal distribution, whose parameters are the gross building density, road density, average road width, the coefficient of variation of building lot frontage, and the ratio of the number of building lot frontages to the number of buildings; and (3) the values of the coefficient of variation of building lot frontages, and that of the ratio of the number of building lot frontages to that of buildings are approximately equal to 0.60 and 1.19, respectively.

Appendix

A log-normal distribution can be derived as a limiting form of the generalised gamma distribution as given by Eq. (1). In Eq. (1), by substituting the following equations into b and c, respectively,

$$ b = \mu_{x} (k\sigma_{x} )^{{\frac{2}{k}}} ,\quad c = \frac{1}{{(k\sigma_{x} )^{2} }}, $$

(17)

we obtain the following equation:

$$ f(x) = \frac{1}{x} \cdot \frac{1}{{\Gamma \left\{ {\frac{1}{{(k\sigma_{x} )^{2} }}} \right\}}} \cdot \frac{k}{{\mu_{x} (k\sigma_{x} )^{{\frac{2}{k}}} }}\left\{ {\frac{1}{{\mu_{x} (k\sigma_{x} )^{{\frac{2}{k}}} }}} \right\}^{{\frac{1}{{k\sigma_{x}^{2} }} - 1}} \exp \left[ {\frac{1}{{k\sigma_{x}^{2} }}\ln x - \frac{1}{{k^{2} \sigma_{x}^{2} }}e^{{k\ln \left( {\frac{x}{{\mu_{x} }}} \right)}} } \right]. $$

(18)

By considering the Taylor expansion of exp[kln(x/μ_x)] in terms of k,

$$ e^{{k\ln \left( {\frac{x}{{\mu_{x} }}} \right)}} = 1 + k\ln \left( {\frac{x}{{\mu_{x} }}} \right) + \frac{1}{2}\left\{ {k\ln \left( {\frac{x}{{\mu_{x} }}} \right)} \right\}^{2} + \sum\limits_{j \ge 3}^{\infty } {\frac{1}{j!}} \left\{ {k\ln \left( {\frac{x}{{\mu_{x} }}} \right)} \right\}^{j} . $$

(19)

Equation (18) is rearranged as follows:

$$ f(x) = \frac{1}{x} \cdot \frac{1}{{\Gamma \left\{ {\frac{1}{{(k\sigma_{x} )^{2} }}} \right\}}}\left\{ {\frac{1}{{(k\sigma_{x} )^{{\frac{2}{k}}} }}} \right\}^{{\frac{1}{{k\sigma_{x}^{2} }}}} \exp \left[ {\ln k - \frac{1}{{(k\sigma_{x} )^{2} }} - \frac{1}{{2\sigma_{x}^{2} }}\left\{ {\ln \left( {\frac{x}{{\mu_{x} }}} \right)} \right\}^{2} + \sum\limits_{j \ge 3}^{\infty } {\frac{{k^{j - 2} }}{{j!\sigma_{x}^{2} }}} \left\{ {\ln \left( {\frac{x}{{\mu_{x} }}} \right)} \right\}^{j} } \right]. $$

(20)

When k tends to zero, by applying the Stirling’s formula for the gamma function, we can derive a log-normal distribution:

$$ f(x) = \frac{1}{{x\sqrt {2\pi \sigma_{x}^{2} } }}e^{{ - \frac{1}{{2\sigma_{x}^{2} }}\left\{ {\ln \left( {\frac{x}{{\mu_{x} }}} \right)} \right\}^{2} }} . $$

(21)